text:
\section{Introduction}

In this work we present a reformulation of the inverse problem for
time dependent systems of second order ordinary differential
equations in terms of semi-basic 1-forms. In this approach we
solely make use of the Fr\"olicher-Nijenhuis theory on
$\pi_{10}:\, J^{1}\pi\rightarrow M$, the first jet bundle of an
$(n+1)$-dimensional, real, smooth manifold $M$, which is fibred
over $\mathbb{R}$, $\pi:M\rightarrow\mathbb{R}$. We characterize
when the time dependent system of SODE
\begin{eqnarray}
\frac{d^{2}x^{i}}{dt^{2}}+2G^{i}\left(t,x,\frac{dx}{dt}\right)=0
\label{sode}
\end{eqnarray}
is equivalent to the system of Euler-Lagrange equations
\begin{eqnarray}
\frac{d}{dt}\left(\frac{\partial L}{\partial
y^{i}}\right)-\frac{\partial L}{\partial x^{i}}=0,\quad
y^{i}=\frac{dx^{i}}{dt}, \label{el_equations}
\end{eqnarray}
for some smooth Lagrangian function $L$ on $J^{1}\pi$, in terms of
a class $\Lambda^1_S(J^1\pi)$ of semi-basic $1$-forms on $J^1\pi$.
This work is a natural extension of the time independent case
studied in \cite{bucataru09}. However, the time dependent
framework has particular aspects that result into some differences
of this approach from the one studied in \cite{bucataru09}.
Lagrangian systems of time independent differential equations are
always conservative and this is not the case in the time dependent
framework. In this approach we use the formalism developed for the
inverse problem of the calculus of variations to search for
symmetries as well. In other words, this approach gives the
possibility of searching for dual symmetries for the time
dependent system \eqref{sode} of SODE in the considered class
$\Lambda^1_S(J^1\pi)$ of semi-basic $1$-forms.

Necessary and sufficient conditions under which the two systems
\eqref{sode} and \eqref{el_equations} can be identified using a
multiplier matrix are usually known as the \emph{Helmholtz
conditions}. This inverse problem was entirely solved only for the
case $n=1$ by Darboux, in 1894, and for $n=2$ by Douglas, in 1941
\cite{douglas41}. A geometric reformulation of Douglas approach,
using linear connections arising from a system of SODE and its
associated geometric structures, can be found in \cite{crampin94,
sarlet02}. The relation between the inverse problem of the
calculus of variations and the condition of self-adjointness for
the equations of variation of the system \eqref{sode} was studied
by Davis in 1929, \cite{davis29}. In 1935, Kosambi
\cite{kosambi35}, has obtained necessary and sufficient
conditions, that were called latter Helmholtz conditions, for the
equations of variations of the system \eqref{sode} to be
self-adjoint. For various approaches to derive the Helmholtz
conditions in both autonomous and nonautonomous case, we refer to
Santilli \cite{santilli78}, Crampin \cite{crampin81}, Henneaux
\cite{henneaux82}, Sarlet \cite{sarlet82}, Marmo et al.
\cite{marmo89}, Morandi et al. \cite{morandi90}, Anderson and
Thompson \cite{anderson92}, Krupkov\'a and Prince
\cite{krupkova08}. See also \cite{bucataru09} for a reformulation
of the Helmholtz conditions in terms of semi-basic $1$-forms in
the time independent case.

In this paper we study the inverse problem of the calculus of
variations when the time dependent system \eqref{sode} of SODE is
identified with a semispray $S$ on the first jet bundle $J^1\pi$.
We seek for a solution of the inverse problem of the calculus of
variations in terms of semi-basic $1$-forms on $J^1\pi$. We first
show, in Theorem \ref{LVF}, that a semispray $S$ is a Lagrangian
vector field if and only if there exists a class of semi-basic
1-forms $\theta$ on $J^{1}\pi$ such that their Lie derivatives
$\mathcal{L}_{S}\theta$ are closed $1$-forms. We denote this class
by $\Lambda^1_S(J^1\pi)$. These results reformulate, in terms of a
semi-basic $1$-form, the results expressed in terms of a $2$-form,
obtained in \cite{anderson92, balachandran80, crampin81,
crampin84a, marmo89}. In Proposition \ref{prop:LVF} we strengthen
the results of Theorem \ref{LVF} and prove that, if nonempty, the
set of semi-basic $1$-forms $\Lambda^1_S(J^1\pi)$ always contains
the Poincar\'e-Cartan 1-form of some Lagrangian function $L$.
Moreover, we show that any semi-basic $1$-form $\theta \in
\Lambda^1_S(J^1\pi)$, which is not the Poincar\'e-Cartan 1-form of
some Lagrangian function, determines a \emph{first integral} and a
\emph{dual symmetry} of the Lagrangian system.

In Theorems \ref{thm:LST1} and \ref{thm:LST2} we characterize the
semi-basic $1$-forms of the set $\Lambda^1_S(J^1\pi)$, depending
if they represent or not Poincar\'e-Cartan 1-forms. First, we pay
attention to $d_J$-closed, semi-basic $1$-forms $\theta$, where
$J$ is the vertical endomorphism. This class of semi-basic
$1$-forms $\theta$ coincides with the class of Poincar\'e-Cartan
$1$-forms corresponding to the Lagrangian function
$L=i_{S}\theta$, see Lemma \ref{LemPoincare2}. For this class we
prove, in Theorem \ref{thm:LST1}, that the inverse problem has a
solution if and only if the Poincar\'e-Cartan $1$-form is
$d_h$-closed, where $h$ is the horizontal projector induced by the
semispray.

In Theorem \ref{thm:LST2} we formulate a coordinate free version
of the Helmholtz conditions in terms of semi-basic $1$-forms on
the first jet bundle $J^1\pi$. If there exists a semi-basic
$1$-form $\theta$ that satisfies the Helmholtz conditions, then
$\theta$ is equivalent (modulo $dt$) with the Poincar\'e-Cartan
$1$-form of some Lagrangian function. Moreover, if such $\theta$
is not $d_J$-closed then $i_Sd\theta$ is a dual-symmetry and
induces a first integral for the semispray $S$. This theorem gives
a characterization for \emph{conservative}, Lagrangian, time
dependent vector fields. To derive the Helmholtz conditions in
Theorem \ref{thm:LST2} we make use of the Fr\"olicher-Nijenhuis
theory on $J^1\pi$ developed in Section \ref{FN-theory}, as well
as of the geometric objects induced by a semispray that are
presented in Section \ref{gobjects}.

Since semi-basic $1$-forms play a key role in our work, we pay a
special attention to this topic in Section \ref{semibasic}. In
this section we prove a Poincar\'e-type Lemma for the differential
operator $d_J$ restricted to the class of semi-basic $1$-forms on
$J^1\pi$. The Poincar\'e-type Lemma will be very useful to
characterize those semi-basic $1$ forms that are the
Poicar\'e-Cartan $1$-forms of some Lagrangian functions.

An important tool in this work is discussed in Section \ref{dcv},
it is a tensor derivation on $J^1\pi$ that is called the dynamical
covariant derivative induced by a semispray. This derivation has
its origins in the work of Kosambi \cite{kosambi35}, where it has
been introduced with the name of "bi-derivative". The notion of
dynamical covariant derivative was introduced by Cari\~nena and
Martinez in \cite{carinena91} as a covariant derivative along the
bundle projection $\pi_{10}$. See also \cite{cantrijn96,
crampin96, jerie02, mestdag01, sarlet95}. It can be defined also
as a derivation on the total space $TJ^1\pi$, see \cite{massa94,
morando95}.

\section{Preliminaries} \label{preliminaries}

\subsection{The first jet bundle $J^{1}\pi$}

For a geometric study of time dependent systems of SODE the most
suitable framework is the affine jet bundle
$(J^{1}\pi,\pi_{10},M)$, see \cite{crampin96, krupkova97, massa94,
saunders89}. We consider an $(n+1)$-dimensional, real and smooth
manifold $M$, which is fibred over $\mathbb{R}$,
$\pi:M\rightarrow\mathbb{R}$. The first jet bundle of $\pi$ is
denoted by $\pi_{10}: J^{1}\pi\to M$,
$\pi_{10}(j_{t}^{1}\gamma)=\gamma(t),$ for $\gamma$ a local
section of $\pi$ and $j_{t}^{1}\gamma$ the first jet of $\gamma$
at $t$. A local coordinate system $(t,x^{i})_{i\in\overline{1,n}}$
on $M$, where $t$ represents the global coordinate on $\mathbb{R}$
and $(x^{i})$ the $\pi$-fiber coordinates, induces a local
coordinate system on $J^{1}\pi,$ denoted by $(t,x^{i},y^{i}).$
Submersion $\pi_{10}$ induces a \emph{natural foliation} on
$J^1\pi$. Coordinates $(t, x^i)$ are transverse coordinates for
this foliation, while $(y^i)$ are coordinates for the leafs of the
foliation.

Throughout the paper we assume that all objects are
$C^{\infty}$-smooth where defined. The ring of smooth functions on
$J^{1}\pi$, the $C^{\infty}$ module of vector fields on $J^{1}\pi$
and the $C^{\infty}$ module of $k$-forms are respectively denoted
by $C^{\infty}(J^{1}\pi)$, ${\mathfrak X}(J^{1}\pi)$ and
$\Lambda^{k}(J^{1}\pi)$. The $C^{\infty}$ module of $(r,s)$-type
tensor fields on $J^{1}\pi$ is denoted by
$\mathcal{T}_{s}^{r}(J^{1}\pi)$ and $\mathcal{T}(J^{1}\pi)$
denotes the tensor algebra on $J^{1}\pi$. By a vector valued
$l$-form ($l\geq0$) on $J^{1}\pi$ we mean an $(1,l)$-type tensor
field on $J^{1}\pi$ that is skew-symmetric in its $l$ arguments.

A parameterized curve on $M$ is a section of $\pi$: $\gamma:
\mathbb{R}\rightarrow M$, $\gamma(t)=(t,x^{i}(t))$. Its
\emph{first jet prolongation} $J^{1}\gamma:
t\in\mathbb{R}\rightarrow J^{1}\gamma(t)=\left(t,x^{i}(t),
{dx^{i}}/{dt}\right)\in J^{1}\pi$ is a section of the fibration
$\pi_{1}:=\pi\circ\pi_{10}:J^{1}\pi\rightarrow\mathbb{R}$.

Let $VJ^{1}\pi$ be the \emph{vertical subbundle} of $TJ^{1}\pi,$
$VJ^{1}\pi=\{\xi\in TJ^{1}\pi,\,\, D\pi_{10}(\xi)=0\}\subset
TJ^{1}\pi$. Its fibers, $V_{u}J^{1}\pi = \operatorname{Ker}
D_{u}\pi_{10}$, $u\in J^{1}\pi$, determine a regular,
$n$-dimensional \emph{vertical distribution}. The vertical
distribution is integrable, being tangent to the natural
foliation. Moreover, $VJ^{1}\pi=spann\{{\partial}/{\partial
y^{i}}\}$ and its annihilators are the \emph{contact 1-forms}
\begin{equation} \delta x^{i}=dx^{i}-y^{i}dt, \quad
i\in\{1,...,n\}. \label{eq:13}\end{equation} One can also view the
vertical subbundle as the image of the \emph{vertical
endomorphism} (or tangent structure)
\begin{equation} J=\frac{\partial}{\partial
y^{i}}\otimes\delta x^{i}. \label{eq:J}\end{equation} The vertical
endomorphism $J$ is a $(1,1)$-type tensor field, and hence a
vector valued 1-form on $J^{1}\pi$, with
$\operatorname{Im}J=VJ^{1}\pi$, $VJ^{1}\pi \subset
\operatorname{Ker}J$ and $J^{2}=0$.

A system \eqref{sode} of second order ordinary differential
equations, whose coefficients depend explicitly on time, can be
identified with a special vector field on $J^{1}\pi$, called a
semispray. A \emph{semispray} is a globally defined vector field
$S$ on $J^{1}\pi$ such that
\begin{equation} J(S)=0 \quad \textrm{and} \quad dt(S)=1. \label{eq:4}\end{equation}
Therefore, a semispray is a vector field $S$ on $J^{1}\pi$
characterized by the property that its integral curves are the
first jet prolongations of sections of $\pi_{1}:\,
J^{1}\pi\rightarrow\mathbb{R}$ . Locally, a semispray has the form
\begin{equation} S=\frac{\partial}{\partial t}+y^{i}\frac{\partial}{\partial
x^{i}}-2G^{i}(t,x,y)\frac{\partial}{\partial
y^{i}},\label{eq:5}\end{equation} where functions $G^{i}$, called
the \emph{semispray coefficients}, are locally defined on
$J^1\pi$.

A parameterized curve $\gamma: I\rightarrow M$ is called a
\emph{geodesic} of $S$ if $S\circ
J^{1}\gamma=\frac{d}{dt}(J^{1}\gamma).$ In local coordinates,
$\gamma(t)=(t,x^{i}(t))$ is a geodesic for the semispray $S$ given
by \eqref{eq:5} if and only if it satisfies the time dependent
system \eqref{sode} of SODE.

\subsection{Fr\"olicher-Nijenhuis theory on $J^{1}\pi$} \label{FN-theory}

In this section we give a short review of the
Fr\"olicher-Nijenhuis theory on $J^{1}\pi$, following
\cite{frolicher56, grifone00, KMS93, deleon89}.

Suppose that $A$ is a vector valued $l$-form on $J^{1}\pi$ and $B$
a (vector valued) $k$-form on $J^{1}\pi$. We can define an
\emph{algebraic derivation} \cite{KMS93} denoted by $i_{A}B$:
\begin{eqnarray*} \label{iaalpha} & & i_AB(X_1,\cdots, X_{k+l-1})=  \\
\nonumber &  & \frac{1}{l!(k-1)!}\sum_{\sigma\in S_{k+l-1}}
\operatorname{sign}(\sigma)\ B\left(A(X_{\sigma(1)},
\cdots,X_{\sigma(l)}), X_{\sigma(l+1)},\cdots, X_{\sigma(k+l-1)}
\right), \end{eqnarray*} where
$X_{1},\cdots,X_{k+l-1}\in{\mathfrak X}(J^{1}\pi)$ and $S_{p}$ is
the permutation group of elements $1,\cdots,p$ . We observe that
$i_{A}B$ is a $(k+l-1)$- form (vector valued form, respectively if
$B$ is vector valued) on $J^{1}\pi$. In \cite{frolicher56,
grifone00} this algebraic derivation is denoted by $B\barwedge A$
and it is called \emph{exterior inner product.}  For $B$ a scalar
form and $l=0$, $A$ is a vector field on $J^{1}\pi$ and $i_{A}B$
is the usual inner product of $k$-form $B$ with respect to vector
field $A$. When $l=1$, then $A$ is a $(1,1)$-type tensor field and
$i_{A}B$ is the $k$-form (or vector valued $k$-form, if $B$ is
vector valued)
\begin{eqnarray}
i_{A}B(X_{1},\cdots,X_{k}) & = & \sum_{i=1}^{k}B(X_{1},\cdots,\,
AX_{i},\cdots,\, X_{k}).\label{eq:inpr11}\end{eqnarray} For any
vector valued $l$-form $A$ on $J^{1}\pi$ we define $i_{A}B=0$, if
$k=0$, which means that $B\in C^{\infty}(J^{1}\pi)$ or
$B\in{\mathfrak X}(J^{1}\pi)$. If $k=1$, we have $i_{A}B=B\circ
A$.

Let $A$ be a vector valued $l$-for$m$ on $J^{1}\pi$, $l\geq0$. The
\emph{exterior derivative} with respect to $A$ is the map
$d_{A}:\Lambda^{k}(J^{1}\pi)\rightarrow\Lambda^{k+l}(J^{1}\pi),\,
k\geq0,$
\begin{eqnarray}
d_{A} & = & i_{A}\circ d-(-1)^{l-1}d\circ
i_{A}.\label{eq:36}\end{eqnarray} In \cite{KMS93}, the exterior
derivative $d_A$ is called the Lie derivative with respect to $A$
and it is denoted by $\mathcal{L}_{A}$. When $A\in{\mathfrak
X}(J^{1}\pi)$ and $k\geq0,$ we obtain $d_{A}=\mathcal{L}_{A}$, the
usual Lie derivative. In this case equation (\ref{eq:36}) is the
well known Cartan's formula. If $A=\operatorname{Id}$, the
identity $(1,1)$-type tensor field on $J^{1}\pi$, then
$d_{\operatorname{Id}}=d$, since
$i_{\operatorname{Id}}\alpha=k\alpha$ for
$\alpha\in\Lambda^{k}(J^{1}\pi)$.

Suppose $A$ and $B$ are vector valued forms on $J^{1}\pi$ of
degrees $l\geq0$ and $k\geq0$, respectively. Then, the
\emph{Fr\"olicher-Nijenhuis bracket }of $A$ and $B$ is the unique
vector valued $(k+l)$-form on $J^{1}\pi$ such that
\cite{frolicher56}
\begin{eqnarray} d_{[A,B]} & = & d_{A}\circ
d_{B}-(-1)^{kl}d_{B}\circ d_{A}.\label{eq:bracket}\end{eqnarray}
When $A$ and $B$ are vector fields, the Fr\"olicher-Nijenhuis
bracket coincides with the usual Lie bracket
$[A,B]=\mathcal{L}_{A}B.$

For a vector field $X\in{\mathfrak X}(J^{1}\pi)$ and a
$(1,1)$-type tensor field $A$ on $J^{1}\pi$, the
Fr\"olicher-Nijenhuis bracket $[X,A]=\mathcal{L}_{X}A$ is the
$(1,1)$-type tensor field \begin{equation}
\mathcal{L}_{X}A=\mathcal{L}_{X}\circ
A-A\circ\mathcal{L}_{X}.\label{eq:LXA}\end{equation}

The Fr\"olicher-Nijenhuis bracket of two $(1,1)$-type tensor
fields $A,\, B$ on $J^{1}\pi$ is the unique vector valued $2$-form
$[A,B]$ defined by \cite{KMS93}
\begin{eqnarray}
[A,B](X,Y) & = & [AX,BY]+[BX,AY]+(A\circ B+B\circ A)[X,Y]\nonumber \\
 &  & -A[X,BY]-A[BX,Y]-B[X,AY]\label{eq:br}\\
 &  & -B[AX,Y],\,\,\forall X,Y\in{\mathfrak X}(J^{1}\pi).\nonumber \end{eqnarray}
In particular, \begin{equation}
\frac{1}{2}[A,A](X,Y)=[AX,AY]+A^{2}[X,Y]-A[X,AY]-A[AX,Y].\label{eq:Nij}\end{equation}
 For a $(1,1)$-type tensor field $A$, the vector valued $2$-form
$N_{A}=\frac{1}{2}[A,A]$ is called the \emph{Nijenhuis tensor} of
$A$.

For the next commutation formulae on $\Lambda^{k}(J^{1}\pi)$,
$k\geq 0$, that will be used throughout the paper, we refer to
\cite[chapter 2]{grifone00}.
\begin{eqnarray}
i_{A}d_{B}-d_{B}i_{A} & = & d_{B\circ A}-i_{[A,B]},\label{eq:com1}\\
\mathcal{L}_{X}i_{A}-i_{A}\mathcal{L}_{X} & = & i_{[X,A]},\label{eq:com2}\\
i_{X}d_{A}+d_{A}i_{X} & = & \mathcal{L}_{AX}-i_{[X,A]},\label{eq:com3}\\
i_{A}i_{B}-i_{B}i_{A} & = & i_{B\circ A}-i_{A\circ
B},\label{eq:com4}\end{eqnarray} for $X\in{\mathfrak X}(J^{1}\pi)$
and $A, B\in\mathcal{T}_{1}^{1}(J^{1}\pi)$.

In this work we will also use the algebraic operator $A^{*}$,
\[ A^{*}B(X_{1},\cdots,X_{k})=B(AX_{1},\cdots,AX_{k}),\] for $A$ a
$(1,1)$-type tensor field and $B$ a (vector valued) $k$-form on
$J^1\pi$.

\subsection{Poincar\'e-type Lemma for semi-basic forms}
\label{semibasic}

For a vector valued $l$-form $A$, we say that a $k$-form $\omega$
on $J^{1}\pi$ is $d_{A}$-\emph{closed} if $d_{A}\omega=0$ and
$d_{A}$-\emph{exact} if there exists
$\theta\in\Lambda^{k-l}(J^{1}\pi)$ such that $\omega=d_{A}\theta.$
For a vector valued $1$-form $A$, from formulae \eqref{eq:bracket}
and \eqref{eq:Nij} we obtain that $d_A^2=d_{N_A}$. Therefore, if
$A$ is not integrable, which means that $N_A\neq 0$, $d_A$-exact
forms may not be $d_A$-closed.

Two forms  $\omega_{1}$ and $\omega_{2}$ on $J^{1}\pi$ are called
\emph{equivalent} (modulo $dt$) if $\omega_{1}\wedge
dt=\omega_{2}\wedge dt$.

For a vector valued $l$-form $A$, we say that a $k$-form
$\theta\in\Lambda^{k}(J^{1}\pi)$ is $d_{A}$-\emph{closed} (modulo
$dt$) if $d_{A}\theta\wedge dt=0$ and $d_{A}$-\emph{exact} (modulo
$dt$) if there exists $\omega$ in $\Lambda^{k-l}(J^{1}\pi)$ such
that $\theta\wedge dt=d_{A}\omega\wedge dt$.

For the vertical endomorphism $J$, its Fr\"olicher-Nijenhuis
tensor is given by
\begin{equation} N_{J}=-\frac{\partial}{\partial
y^{i}}\otimes\delta x^{i}\wedge dt=-J\wedge
dt.\label{eq:NijenhuisJ}\end{equation} Consequently, using formula
\eqref{eq:bracket}, we obtain $d_{J}^{2}=d_{N_J}=-d_{J\wedge
dt}\neq 0$. Therefore, $d_{J}$-exact forms on $J^{1}\pi$ may not
be $d_{J}$-closed. However, we will prove in the first part of
Lemma \ref{lem:(Poincare)} that $d_{J}$-exact (modulo $dt$) forms
are $d_{J}$-closed (modulo $dt$) forms. In the second part of
Lemma \ref{lem:(Poincare)} we will show that the converse is true
for semi-basic $k$-forms, only.

For a $k$-form $\omega$, the following identities can be obtained
immediately
\begin{eqnarray}
\nonumber  i_{J\wedge dt} \omega= (-1)^{k+1} i_J\omega \wedge dt, \\
\label{djdt} d_{J\wedge dt} \omega= (-1)^{k+1} d_J\omega \wedge
dt.
\end{eqnarray}

\begin{defn}
i) A $k$-form $\omega$ on $J^{1}\pi$, $k\geq1$, is called
\emph{semi-basic} if $\omega(X_{1},\cdots,X_{k})$$=0$, when one of
the vectors $X_{i}$, $i\in\{1,...,k\}$, is vertical.

ii) A vector valued $k$-form $A$ on $J^{1}\pi$ is called
\emph{semi-basic} if it takes values in the vertical subbundle and
$A(X_{1},\cdots,X_{k})=0$, when one of the vectors $X_{i}$, $
i\in\{1,...,k\}$ is vertical.
\end{defn}
A semi-basic $k$-form verifies the relation $i_{J}\theta=0$. The
converse is true only for $k=1$. Semi-basic $1$-forms are
annihilators for the vertical distribution and hence locally can
be expressed as
$\theta=\theta_{0}(t,x,y)dt+\theta_{i}(t,x,y)\delta x^{i}$.
Contact 1-forms $\delta x^{i}$ given by formula \eqref{eq:13} are
semi-basic 1-forms.

\begin{defn} \label{def:regtheta}
A semi-basic $1$-form $\theta$ on $J^1\pi$ is called
\emph{non-degenerate} if the $2$-form $d\theta+i_Sd\theta\wedge
dt$ has rank $2n$ on $J^1\pi$.
\end{defn}

If a vector valued $k$-form $A$ is semi-basic, then $J\circ A=0$,
$i_{J}A=0$ and $J^{*}A=0$. The converse is true only for $k=1$. It
follows that the vertical endomorphism $J$ is a vector valued,
semi-basic $1$-form.

Next lemma presents a characterization of forms that are
equivalent (modulo $dt$) to the null form and will be used to
prove a Poincar\'e-type Lemma for semi-basic forms.

\begin{lem} \label{omegadt}
 A $k$-form $\omega$ on $J^{1}\pi$ satisfies the condition
   $\omega\wedge dt=0$ if and only if it is of the form $\omega=i_{S\otimes dt}\omega=(-1)^{k+1}i_{S}\omega\wedge dt$,
   for an arbitrary semispray $S$.
\end{lem}
\begin{proof}
A simple computation gives $i_{S\otimes
dt}\omega=(-1)^{k+1}i_{S}\omega\wedge dt$. Therefore, if
$\omega=i_{S\otimes dt}\omega=(-1)^{k+1}i_{S}\omega\wedge dt$ it
follows that $\omega\wedge dt=0$. Conversely, if $\omega\wedge
dt=0$, we apply $i_{S}$ to this identity and get
$0=i_{S}\omega\wedge dt+(-1)^{k}\omega$. Hence,
$\omega=(-1)^{k+1}i_{S}\omega\wedge dt=i_{S\otimes dt}\omega$,
which completes the proof.
\end{proof}
\begin{lem}
\label{lem:(Poincare)}(Poincar\'e-type Lemma) Consider $\theta$ a
$k$-form on $J^{1}\pi$. \begin{itemize}
\item[i)] If $\theta$ is $d_J$-exact (modulo $dt$) then $\theta$
is $d_J$-closed (modulo $dt$).
\item[ii)] If $\theta$ is a semi-basic form and $d_{J}$-closed
(modulo $dt$), then $\theta$ is locally $d_{J}$-exact (modulo
$dt$).
\end{itemize}
\end{lem}
\begin{proof}
i) Suppose that $\theta\in\Lambda^{k}(J^{1}\pi)$ is $d_{J}$-exact
(modulo $dt$). Then, there exists
$\omega\in\Lambda^{k-1}(J^{1}\pi)$ such that $\theta\wedge dt
=d_{J}\omega\wedge dt$. If we apply $d_J$ to both sides of this
identity, and use formula \eqref{djdt}, we obtain
\begin{eqnarray*}
d_{J}\theta\wedge dt = d_{J}^{2}\omega\wedge dt = -d_{J\wedge
dt}\omega \wedge dt =0. \end{eqnarray*} which means that the
$k$-form $\theta$ is $d_J$-closed (modulo $dt$).

ii) We have to prove that for $\theta\in\Lambda^{k}(J^{1}\pi)$
semi-basic, with $d_{J}\theta\wedge dt=0$, there exists
$\omega\in\Lambda^{k-1}(J^{1}\pi)$ (locally defined), such that
$\theta\wedge dt=d_{J}\omega\wedge dt$. Since $\theta$ is
semi-basic, it follows that locally it has the form
\begin{eqnarray*}
\theta=\frac{1}{k!}\theta_{i_1...i_k}\delta x^{i_1}\wedge \cdots
\wedge \delta x^{i_k} +
\frac{1}{(k-1)!}\widetilde{\theta}_{i_1...i_{k-1}}\delta
x^{i_1}\wedge \cdots \wedge \delta x^{i_{k-1}}\wedge dt.
\end{eqnarray*}
Using the identities $ i_Jdt=0$, $i_J\delta x^i=0$, and
$i_Jdy^i=\delta x^i$, we obtain
\begin{eqnarray*}
d_J\theta\wedge dt = i_Jd\theta \wedge dt = \frac{1}{k!}
\frac{1}{(k+1)!} \varepsilon^{jj_1...j_k}_{i_1i_2...i_{k+1}}
\frac{\partial \theta_{j_1...j_k}}{\partial y^j} \delta
x^{i_1}\wedge \cdots \wedge \delta x^{i_{k+1}}\wedge dt,
\end{eqnarray*}
where $\varepsilon^{jj_1...j_k}_{i_1i_2...i_{k+1}}$ is Kronecker's
symbol. Hence
\begin{eqnarray*}
\frac{1}{(k+1)!} \varepsilon^{jj_1...j_k}_{i_1i_2...i_{k+1}}
\frac{\partial \theta_{j_1...j_k}}{\partial y^j}=0.
\end{eqnarray*} If one considers the transverse coordinates
$t,x^i$ of the natural foliation as parameters, one can use
Poincar\'e Lemma on the leafs of this foliation to obtain (locally
defined) functions $\omega_{j_i...j_{k-1}}(t,x^i,y^i)$ such that
\begin{eqnarray}
\theta_{i_1...i_k}=\frac{1}{(k-1)!}
\varepsilon^{jj_1...j_{k-1}}_{i_1i_2...i_{k}} \frac{\partial
\omega_{j_1...j_{k-1}}}{\partial y^j}. \label{tomega}
\end{eqnarray} Therefore, one can consider the (locally defined) semi-basic $(k-1)$-form
\begin{eqnarray*}
\omega=\frac{1}{(k-1)!}\omega_{i_1...i_{k-1}}\delta x^{i_1}\wedge
\cdots \wedge \delta x^{i_{k-1}}. \end{eqnarray*} In view of
identity \eqref{tomega} we have $\theta\wedge dt = d_J\omega
\wedge dt$, which means that $\theta$ is $d_J$-exact (modulo
$dt$).
\end{proof}

For semi-basic forms, the differential operator $d_{J}$ is closely
related to the exterior differential $d''$ along the leafs of the
natural foliation, studied by Vaisman in \cite{vaisman71}. This
relation as well as the proof of Theorem 3.1 from \cite{vaisman71}
can be used to give a different proof for Lemma
\ref{lem:(Poincare)}. This proof will require to fix a
distribution supplementary to the vertical distribution, which is
always possible, see Section \ref{nonlinear_connection}.

For semi-basic $1$ and $2$-forms a Poincar\'e-type Lemma and its
usefulness for the inverse problem of the calculus of variations
is discussed in Marmo et al. \cite{marmo89}. For forms along the
tangent bundle projection a similar result as in Lemma
\ref{lem:(Poincare)} has been shown in \cite[Prop. 2.1]{sarlet95}.

In Section \ref{section_lvf} we will use Poincar\'e-type Lemma
\ref{lem:(Poincare)} to find necessary and sufficient conditions
for a semi-basic $1$-form on $J^{1}\pi$ to coincide, or to be
equivalent (modulo $dt$), with the Poincar\'e-Cartan 1-form of
some Lagrangian function.

\section{Geometric objects induced by a semispray}
\label{gobjects}

In this section we present some geometric structures that can de
derived from a semispray using the Fr\"olicher-Nijenhuis theory:
nonlinear connection, Jacobi endomorphism, dynamical covariant
derivative. See \cite{cantrijn96, crampin96, deleon88, deleon89,
miron94, sarlet95}. These structures will be used later to express
necessary and sufficient conditions for a given semispray to be a
Lagrangian vector field.

\subsection{Nonlinear connection} \label{nonlinear_connection}

A \emph{nonlinear connection} on the first jet bundle $J^{1}\pi$
is an $(n+1)$-dimensional distribution $H:u\in J^1\pi \mapsto
H_u\subset T_uJ^1\pi$, supplementary to the vertical distribution
$VJ^{1}\pi$. This means that for each $u \in J^1\pi$, we have the
direct decomposition $T_uJ^1\pi=H_u\oplus V_u$.

A semispray $S$ induces a nonlinear connection on $J^{1}\pi$,
determined by the \emph{almost product structure}, or dynamical
connection \cite{deleon88}
\begin{equation} \Gamma=-\mathcal{L}_{S}J+S\otimes
dt, \quad \Gamma^{2}=\operatorname{Id}. \label{eq:6}\end{equation}
The \emph{horizontal projector} that corresponds to this almost
product structure is \begin{equation}
h=\frac{1}{2}\left(\operatorname{Id} - \mathcal{L}_{S}J+S\otimes
dt\right)\label{eq:7}\end{equation} and the \emph{vertical
projector} is $v=\operatorname{Id}-h.$ Note that in this paper we
chose to work with \emph{the weak horizontal projector} $h$
defined by formula \eqref{eq:7}. We can also consider
$h_{0}=S\otimes dt$ and $h_{1}=h-h_{0}$, the \emph{strong
horizontal projector}. The horizontal subspace is the eigenspace
corresponding to the eigenvalue $+1$ of $\Gamma,$ while the
vertical subspace is the eigenspace corresponding to the
eigenvalue $-1$. The horizontal subspace is spanned by $S$ and by
$$ \frac{\delta}{\delta x^{i}}=\frac{\partial}{\partial
x^{i}}-N_{i}^{j}\frac{\partial}{\partial y^{j}}, \quad
\textrm{where}  \quad N_{j}^{i}=\frac{\partial G^{i}}{\partial
y^{j}}.$$ From now on, whenever a semispray will be given, we
prefer to work with the following adapted basis and cobasis
\begin{equation}
\left\{ S,\frac{\delta}{\delta x^{i}},\frac{\partial}{\partial
y^{i}}\right\} , \quad \{dt,\delta x^{i},\delta y^{i}\},
\label{eq:11}\end{equation} with $\delta x^{i}=dx^{i}-y^{i}dt$ the
\emph{contact 1-forms} and
\begin{equation} \delta
y^{i}=dy^{i}+N_{j}^{i}dx^{j}+N_{0}^{i}dt,\quad
N_{0}^{i}=2G^{i}-N_{j}^{i}y^{j}.\label{eq:14}\end{equation}
Functions $N_{j}^{i}$ and $N_{0}^{i}$ are the coefficients of the
nonlinear connection induced by the semispray $S$. The $1$-forms
$\delta y^i$ are annihilators for the horizontal distribution.

With respect to the adapted basis and cobasis \eqref{eq:11} the
almost product structure can be locally expressed as
$$ \Gamma=S\otimes dt+\frac{\delta}{\delta x^{i}}\otimes\delta
x^{i}-\frac{\partial}{\partial y^{i}}\otimes\delta y^{i}.$$
Therefore, the horizontal and vertical projectors are locally
expressed as
$$ h=S\otimes dt+\frac{\delta}{\delta
x^{i}}\otimes\delta x^{i}, \quad v=\frac{\partial}{\partial
y^{i}}\otimes\delta y^{i}.$$ We consider the (1,1)-type tensor
field, which corresponds to the almost complex structure in the
autonomous case,
\begin{equation}
\mathbb{F}=h\circ\mathcal{L}_{S}h-J. \label{eq:21}\end{equation}
Tensor $\mathbb{F}$ satisfies $\mathbb{F}^{3}+\mathbb{F}=0,$ which
means that it is an $f(3,1)$ structure. It can be expressed
locally as
$$\mathbb{F}=\frac{\delta}{\delta x^{i}}\otimes\delta
y^{i}-\frac{\partial}{\partial y^{i}}\otimes\delta x^{i}.$$ For
some useful future calculus, we will also need the next formulae
\begin{eqnarray} \left[\frac{\delta}{\delta
x^{i}},\frac{\delta}{\delta x^{j}}\right] & = &
R_{ij}^{k}\frac{\partial}{\partial y^{k}},\quad
R_{jk}^{i}=\frac{\delta N_{j}^{i}}{\delta x^{k}}-\frac{\delta N_{k}^{i}}{\delta x^{j}},\label{eq:20} \\
\left[\frac{\delta}{\delta x^{i}},\frac{\partial}{\partial y^{j}}\right] & = &
\frac{\partial N_{i}^{k}}{\partial y^{j}}\frac{\partial}{\partial y^{k}}=
\frac{\partial^{2}G^{k}}{\partial y^{i}\partial y^{j}}\frac{\partial}{\partial y^{k}},\label{eq:20'}
\end{eqnarray}
for the Lie brackets of the vector fields of the adapted basis
\eqref{eq:11}. We also have:
\begin{eqnarray}
\mathcal{L}_{S}S=0, & \mathcal{L}_{S}
\displaystyle\frac{\delta}{\delta x^{i}}= N_{i}^{j}
\frac{\delta}{\delta x^{j}} + R_{i}^{j} \frac{\partial}{\partial
y^{j}}, & \mathcal{L}_{S}\frac{\partial}{\partial y^{i}} = -
\frac{\delta}{\delta x^{i}} + N_{i}^{j}\frac{\partial}{\partial y^{j}}, \label{eq:LSV}\\
\mathcal{L}_{S}dt=0, & \mathcal{L}_{S}\delta x^{i}=
-N_{j}^{i}\delta x^{j}+\delta y^{i}, & \mathcal{L}_{S}\delta y^{i}
=  - R_{j}^{i}\delta x^{j}-N_{j}^{i}\delta y^{j},
\label{eq:LSCV}\end{eqnarray} where
\begin{equation}
R_{j}^{i}=2\frac{\partial G^{i}}{\partial x^{j}}-\frac{\partial
G^{i}}{\partial y^{k}}\frac{\partial G^{k}}{\partial
y^{j}}-S\left(\frac{\partial G^{i}}{\partial y^{j}}\right).
\label{eq:18}\end{equation} $R_{j}^{i}$ are the components of a
$(1,1)$-type tensor field on $J^{1}\pi$, known as the \emph{second
invariant in KCC-theory} \cite{antonelli01}, \emph{the Douglas
tensor} \cite{douglas41, grifone00} or as the \emph{Jacobi
endomorphism} \cite{cantrijn96, crampin84a, jerie02, sarlet95}.

Using formulae \eqref{eq:J}, \eqref{eq:7} and the set of relations
\eqref{eq:LSV}, \eqref{eq:LSCV}, we obtain
\begin{eqnarray}
\mathcal{L}_{S}J & = & -\frac{\delta}{\delta x^{i}}\otimes\delta x^{i}+\frac{\partial}{\partial y^{i}}\otimes\delta y^{i},\label{eq:23}\\
\mathcal{L}_{S}h & = & \frac{\delta}{\delta x^{i}}\otimes\delta
y^{i}+R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes\delta
x^{i}.\label{eq:24}\end{eqnarray}

According to formulae \eqref{eq:br}, \eqref{eq:Nij},
\eqref{eq:20}, \eqref{eq:20'}, and
$[J,\Gamma]=2[J,h]-[J,\operatorname{Id}]=2[J,h]$ one can prove the
following proposition.
\begin{prop}
i) The weak torsion tensor field of the nonlinear connection
$\Gamma$ vanishes: $[J,h]=0,$ which is equivalent also with
$[J,\Gamma]=0$.

ii) The curvature tensor $R=N_h$ of the nonlinear connection
$\Gamma$ is a vector valued semi-basic 2-form, locally given by
\begin{equation}
R=\frac{1}{2}[h,h]=\frac{1}{2}R_{ij}^{k}\frac{\partial}{\partial
y^{k}}\otimes\delta x^{i}\wedge\delta
x^{j}+R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes
dt\wedge\delta x^{i},\label{eq:19}\end{equation} where
$R_{ij}^{k}$ and $R_{i}^{j}$ are given in formulae \eqref{eq:20}
and \eqref{eq:18}.
\end{prop}
The \emph{Jacobi endomorphism} can be defined as follows.
\begin{equation}
\Phi=v\circ\mathcal{L}_{S}h=\mathcal{L}_{S}h-\mathbb{F}-J.\label{eq:22}\end{equation}
It is a semi-basic, vector valued 1-form and satisfies
$\Phi^{2}=0$. Locally, it can be expressed as
\begin{eqnarray} \Phi=R_{i}^{j}\frac{\partial}{\partial
y^{j}}\otimes\delta x^{i}.\label{localphi}\end{eqnarray} As we can
see from formula \eqref{eq:19}, the Jacobi endomorphism is part of
the curvature tensor $R$, which is the \emph{third invariant in
KCC theory}, \cite{antonelli01}. Moreover, as we will see in the
next proposition, the curvature tensor can be obtained, using
Fr\"olicher-Nijenhuis theory, from the Jacobi endomorphism. The
result is similar to the one from \cite{sarlet95}, where different
techniques are used.

\begin{prop}
The Jacobi endomorphism and the curvature of the nonlinear
connection are related by the following formulae.
\begin{eqnarray}
\Phi & = & i_{S}R, \label{eq:23'} \\
\left[J,\Phi\right] & = & 3R + \Phi\wedge dt. \label{eq:24'}
\end{eqnarray}
\end{prop}
\begin{proof}
Since $2R=[h,h]$, for $X$ in ${\mathfrak X}(J^{1}\pi)$, we have $$
2(i_{S}R)(X) = [h,h](S,X) = 2[S,hX]-2h[S,hX]=2v[S,hX] =
2\Phi(X),$$ which proves formula \eqref{eq:23'}. To prove formula
\eqref{eq:24'} we use the local expression \eqref{eq:J} and
\eqref{localphi} of $J$ and $\Phi$ as well as the identity
$$ \frac{\partial R_{j}^{k}}{\partial y^{i}}-\frac{\partial R_{i}^{k}}{\partial
y^{j}}=3R_{ij}^{k},$$ which follows by a direct computation. The
Fr\"olicher-Nijenhuis bracket of $J$ and $\Phi$ is given by
\begin{eqnarray*}
[J,\Phi] & = & 2R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes
dt\wedge\delta x^{i} +\frac{1}{2}\left(\frac{\partial
R_{j}^{k}}{\partial y^{i}}-
\frac{\partial R_{i}^{k}}{\partial y^{j}}\right)\frac{\partial}{\partial y^{k}}\otimes\delta x^{i}\wedge\delta x^{j}\\
 & = & 2dt\wedge\Phi+\frac{3}{2}R_{ij}^{k}\frac{\partial}{\partial y^{k}}\otimes\delta x^{i}\wedge\delta x^{j}.\end{eqnarray*}
Using the above formula and formula \eqref{eq:19} that defines the
curvature $R$ we obtain formula \eqref{eq:24'}, which completes
the proof. \end{proof}

The following $(1,1)$-type tensor field $\Psi$ will allow us to
express in a simpler form the dynamical covariant derivative
induced by $S$, which will be discussed in the next section.
\begin{eqnarray}
\Psi & = & h\circ\mathcal{L}_{S}h+v\circ\mathcal{L}_{S}v=\Gamma\circ\mathcal{L}_{S}h = \mathbb{F}+J-\Phi,\label{eq:Psi}\\
\Psi & = & \frac{\delta}{\delta x^{i}}\otimes\delta
y^{i}-R_{i}^{j}\frac{\partial}{\partial y^{j}}\otimes\delta
x^{i}.\label{eq:Psilocal}\end{eqnarray}

\subsection{Dynamical covariant derivative} \label{dcv}

For a semispray $S$ there are various possibilities to define a
tensor derivation on $J^{1}\pi$. Such derivation was considered
first by Kosambi in \cite{kosambi35}, with the name of
biderivative. This derivation has been rediscovered latter and was
called the dynamical covariant derivative \cite{carinena91}. It
can be defined either as a derivation along the bundle projection
$\pi_{10}$, as in \cite{cantrijn96, crampin96, jerie02, mestdag01,
sarlet95} or as a derivation on the total space $TJ^1\pi$,
\cite{massa94, morando95}.

In this section we use Fr\"olicher-Nijenhuis theory to define the
dynamical covariant derivative as a tensor derivation on
$J^{1}\pi$, following the time independent case developed in
\cite{bucataru09}. In Proposition
\ref{pro:The-dynamical-covariant} we present some useful
commutation rules of the dynamical covariant derivative with the
geometric structures studied in the previous sections.

A map $\nabla: \mathcal{T}(J^{1}\pi)
\rightarrow\mathcal{T}(J^{1}\pi)$ is a \emph{tensor derivation} on
$\mathcal{T}(J^{1}\pi)$ if it satisfies the following conditions.
\begin{itemize}
\item[i)] $\nabla$ is $\mathbb{R}$-linear.
\item[ii)] $\nabla$ preserves the type of tensor fields.
\item[iii)] $\nabla$ obeys the Leibnitz rule: $\nabla(T\otimes
S)=\nabla T\otimes S+T\otimes\nabla S$, $\forall
T,S\in\mathcal{T}(J^{1}\pi)$.
\item[iv)] $\nabla$ commutes with any contractions.
\end{itemize}

For a semispray $S$ on $J^{1}\pi,$ we define the
$\mathbb{R}$-linear map $\nabla_{0}:{\mathfrak
X}(J^{1}\pi)\rightarrow{\mathfrak X}(J^{1}\pi)$
\begin{equation}
\nabla_{0}X=h[S,hX]+v[S,vX].\label{eq:25}\end{equation} It follows
that $\nabla_{0}(fX)=S(f)X+f\nabla_{0}X$, for all
$f\in\mathcal{C}^{\infty}(J^{1}\pi)$ and $X\in{\mathfrak
X}(J^{1}\pi)$.

Since any tensor derivation on $\mathcal{T}(J^{1}\pi)$ is
completely determined by its action over the smooth functions and
vector fields on $J^{1}\pi$, there exists a unique tensor
derivation $\nabla$ on $J^{1}\pi$ such that
$\nabla\vert_{\mathcal{C}^{\infty}(J^{1}\pi)}=S$ and $
\nabla\vert_{{\mathfrak X}(J^{1}\pi)}=\nabla_{0}$. This tensor
derivation is called the \emph{dynamical covariant derivative}
induced by the semispray $S$.

Next, we will obtain some alternative expressions for the action
of the dynamical covariant derivative $\nabla$ on ${\mathfrak
X}(J^{1}\pi)$, $\Lambda^{k}(J^{1}\pi)$ and $(1,1)$-type tensor
fields on $J^{1}\pi.$

From formula \eqref{eq:25} we obtain
$$\nabla\vert_{{\mathfrak X}(J^{1}\pi)}=h\circ\mathcal{L}_{S}\circ
h+v\circ\mathcal{L}_{S}\circ v = \mathcal{L}_{S} +
h\circ\mathcal{L}_{S}h+v\circ\mathcal{L}_{S}v.$$ Using the
$(1,1)$-type tensor field $\Psi$ defined in formula
\eqref{eq:Psi}, we obtain the following expression of the
dynamical covariant derivative.
\begin{equation} \nabla\vert_{{\mathfrak
X}(J^{1}\pi)}=\mathcal{L}_{S}+\Psi.\label{eq:26}\end{equation}

Since $\nabla$ satisfies the Leibnitz rule, we deduce that the
action of $\nabla$ on $k$-forms is given by
\begin{equation}
\nabla\vert_{\Lambda^{k}(J^{1}\pi)}=\mathcal{L}_{S}-i_{\Psi}.\label{eq:27}\end{equation}
Above formula \eqref{eq:27} implies that $\nabla$ is a degree zero
derivation on $\Lambda^{k}(J^{1}\pi)$. Therefore \cite[p.
69]{KMS93}, it can be uniquely written as a sum of a Lie
derivation, which is $\mathcal{L}_{S}$, and an algebraic
derivation, given by $i_{\Psi}$.

Similarly, we deduce that the action of $\nabla$ on a vector
valued $k$-form $A$ on $J^{1}\pi$ is given by
\begin{equation}
\nabla A = \mathcal{L}_{S}A + \Psi\circ
A-i_{\Psi}A.\label{eq:28}\end{equation}
\begin{prop}
\label{pro:The-dynamical-covariant} The dynamical covariant
derivative induced by a semispray $S$ has the following
properties.

i) $\nabla S=0$ and $\nabla i_{S}=i_{S}\nabla$.

ii) $\nabla h=\nabla v=0,$ which means that $\nabla$ preserves by
parallelism the horizontal and vertical distributions.

iii) $\nabla J=\nabla\mathbb{F}=0,$ which means that $\nabla$ acts
similarly on both horizontal and vertical distributions.

iv) The restriction of $\nabla$ to $\Lambda^{k}(J^{1}\pi)$ and the
exterior differential operator $d$ satisfy the commutation formula
\begin{equation}
d\nabla-\nabla d=d_{\Psi}.\label{eq:30}\end{equation}

v) The restriction of $\nabla$ to $\Lambda^{k}(J^{1}\pi)$
satisfies the commutation rule
\begin{equation} \nabla i_{A}-i_{A}\nabla=i_{\nabla
A}\label{eq:31}\end{equation} for a $(1,1)$- type tensor field $A$
on $J^{1}\pi.$ Hence the algebraic derivations with respect to $h,
v, J, \mathbb{F}$ commute with
$\nabla\vert_{\Lambda^{k}(J^{1}\pi)}.$
\end{prop}
\begin{proof}
First item follows directly using definition formula \eqref{eq:25}
and formula \eqref{eq:27}.

From the definition formula \eqref{eq:Psi} of tensor $\Psi$ we
obtain
\begin{equation} A\circ\Psi-\Psi\circ
A=\mathcal{L}_{S}A,\label{eq:32}\end{equation} for $A\in\{h, v, J,
\mathbb{F}\}.$ Using formula \eqref{eq:28}, it follows immediately
that $\nabla h=\nabla v=\nabla J=\nabla\mathbb{F}=0$ and hence we
proved item ii) and iii) of the proposition.

Using formula \eqref{eq:27} we obtain
\[ d\nabla=d\mathcal{L}_{S}-di_{\Psi}=\mathcal{L}_{S}d-i_{\Psi}d+d_{\Psi}=\nabla
d+d_{\Psi},\] for the restriction of $\nabla$ to
$\Lambda^{k}(J^{1}\pi)$. Therefore, formula \eqref{eq:30} is true.

From formulae \eqref{eq:28}, \eqref{eq:com2}, as well as
\eqref{eq:com4} we obtain
\[ \nabla i_{A}-i_{A}\nabla = \mathcal{L}_{S}i_{A} -
i_{A}\mathcal{L}_{S}-i_{\Psi}i_{A}+i_{A}i_{\Psi} =
i_{[S,A]}-i_{A\circ\Psi}+i_{\Psi\circ A}=i_{\nabla A},\] which
proves the last item of the proposition.
\end{proof}
First three items of Proposition \ref{pro:The-dynamical-covariant}
can be locally expressed as follows.
\begin{eqnarray}
\nabla S=0, &  & \nabla dt=0,\nonumber \\
\nabla\frac{\delta}{\delta x^{i}}=N_{i}^{j}\frac{\delta}{\delta x^{j}}, &  & \nabla\delta x^{i}=-N_{j}^{i}\delta x^{j},\label{eq:nablalocal}\\
\nabla\frac{\partial}{\partial
y^{i}}=N_{i}^{j}\frac{\partial}{\partial y^{j}}, &  & \nabla\delta
y^{i}=-N_{j}^{i}\delta y^{j}.\nonumber \end{eqnarray}

Tensor derivation $\nabla$ coincides with the dynamical covariant
derivative induced by the Berwald linear connection $\hat{\nabla}$
on $J^{1}\pi$, studied by Massa and Pagani in \cite{massa94}, in
the following sense $\nabla=\hat{\nabla}_{S}$. See also
\cite{mestdag01} for a detailed study of Berwald-type connection
associated to time dependent systems of SODE. The relation between
the dynamical covariant derivative and the Berwald connection
implies that tensor $\Psi$ is the shape map $A_{S}$ on the
manifold $N=J^{1}\pi,$ studied by Jerie and Prince in
\cite{jerie02} on an arbitrary manifold.

Another tensor derivation on ${\mathfrak X}(J^{1}\pi)$, induced by
a semispray, was proposed by Morando and Pasquero
\cite{morando95}. In our notations, Morando and Pasquero's
derivation can be expressed as $\nabla+\Psi = \mathcal{L}_S+ 2
\Psi$.

Next result is a technical lemma that expresses the action of the
dynamical covariant derivative on semi basic $1$-forms, and will
be useful for the proof of Theorem \ref{thm:LST2}.
\begin{lem} \label{lemnablatheta} For a semi-basic 1-form $\theta$ on
$J^{1}\pi,$ its dynamical covariant derivative can be expressed as
follows.
\begin{equation}
\nabla\theta=d_{h}i_{S}\theta+i_{S}d_{h}\theta.
\label{eq:nablatheta}\end{equation} Moreover, $\theta$ satisfies
the identity
\begin{equation}
i_{S}d_{h}\theta=i_{h}i_{S}d\theta.\label{eq:isdht}\end{equation}
\end{lem}
\begin{proof}
Using the identity
$d_{h}i_{S}+i_{S}d_{h}=\mathcal{L}_{S}-i_{[S,h]},$ one gets
\begin{equation}
\mathcal{L}_{S}\theta=d_{h}i_{S}\theta+i_{S}d_{h}\theta+i_{[S,h]}\theta.\label{eq:derLie}\end{equation}
Since $\theta$ is semi-basic we have
$i_{[S,h]}\theta=i_{\mathbb{F}+J+\Phi}\theta=i_{\mathbb{F}}\theta.$
Moreover,
$i_{\Psi}\theta=i_{\mathbb{F}+J-\Phi}\theta=i_{\mathbb{F}}\theta$
and hence
$\nabla\theta=\mathcal{L}_{S}\theta-i_{\Psi}\theta=d_{h}i_{S}\theta+i_{S}d_{h}\theta.$

For the second identity, we notice that
$i_{S}d_{h}\theta=i_{S}i_{h}d\theta-i_{S}d\theta.$ Using
$i_{S}i_{h}-i_{h}i_{S}=i_{hS}=i_{S},$ one gets
$i_{S}d_{h}\theta=i_{h}i_{S}d\theta$.\end{proof}

\subsection{Dual symmetries} \label{symmetries}

In Section \ref{sec_Helm} we will search for solutions of the
inverse problem of the calculus of variations in terms of
semi-basic $1$-forms. We prove that in the Lagrangian case there
is always a solution that is a $d_J$-closed semi-basic $1$-form,
which is the Poincar\'e-Cartan $1$-form of some Lagrangian
function. If the solution of the inverse problem contains a
semi-basic $1$-form that is not $d_J$-closed then it induces a
dual symmetry and a first integral of the semispray. For
discussions regarding adjoint symmetries in the context of the
inverse problem of the calculus of variations we refer to
\cite{carinena89, carinena91, sarlet90}. A detailed discussion of
symmetries, dual symmetries, adjoint symmetries and the relations
among them can be found in \cite{morando95}. In this section we
use expression \eqref{eq:27} of the dynamical covariant derivative
to characterize dual symmetries in terms of a Jacobi equation.

\begin{defn}
A 1-form $\omega$ on $J^{1}\pi$ is called a \emph{dual symmetry}
(or an invariant form) for a semispray $S$ if
$\mathcal{L}_{S}\omega=0$.
\end{defn}

In this work we will be interested in dual symmetries for which
$i_{S}\omega=0$. Since we will work with equivalence classes
(modulo $dt$) of dual symmetries, in each class we can choose a
representant of this form.

To express the condition $\mathcal{L}_{S}\omega=0$ for a $1$-form,
locally expressed as $\omega=\tilde{\omega}_{i}\delta
x^{i}+\omega_{i}\delta y^{i}$, we will use formulae \eqref{eq:27}
and \eqref{eq:nablalocal} for the dynamical covariant derivative.
Therefore $\mathcal{L}_S\omega=0$, which is equivalent to $\nabla
\omega=-i_{\Psi}\omega$ can be locally expressed as
\begin{eqnarray*}
\nabla\tilde{\omega}_{i}\delta x^{i}+\nabla\omega_{i}\delta
y^{i}=-\tilde{\omega}^{i}\delta y^{i}+R_{i}^{j}\omega_{j}\delta
x^{i} & \Leftrightarrow & \begin{cases}
\tilde{\omega}_{i} & =-\nabla\omega_{i},\\
\nabla\tilde{\omega}_{i}-R_{i}^{j}\omega_{j} &
=0.\end{cases}\end{eqnarray*} Hence $\omega$ is a dual symmetry,
if and only if $\omega=-\nabla\omega_{i}\delta
x^{i}+\omega_{i}\delta y^{i}$ and it satisfies the \emph{Jacobi
equation}
\begin{equation}
\nabla^{2}\omega_{i}+R_{i}^{j}\omega_{j}=0.
\label{eq:dualsymlocal}\end{equation} If $\omega$ is a dual
symmetry then $-i_{\Gamma}\omega$ is an adjoint symmetry. Locally,
an adjoint symmetry $\alpha$ with $i_{S}\alpha=0$ is locally
expressed as $\alpha=\nabla\omega_{i}\delta x^{i}+\omega_{i}\delta
y^{i}$ and verifies the same equation \eqref{eq:dualsymlocal}.

\section{Lagrangian vector fields}\label{section_lvf}

An approach to the inverse problem of the calculus of variations
seeks for the existence of a non-degenerate multiplier matrix
$g_{ij}(t,x,y)$ which relates the geodesic equations \eqref{sode}
of a semispray with the Euler-Lagrange equations
\eqref{el_equations} for a Lagrangian function $L$. In this case,
the Lagrangian function $L$ is determined from the condition that
the multiplier matrix $g_{ij}$ is the Hessian of $L$. Necessary
and sufficient conditions for the existence of such multiplier
matrix are known as Helmholtz conditions and were obtained, using
various techniques for both autonomous and nonautonomous case, in
\cite{anderson92, bucataru09, crampin81, henneaux82, kosambi35,
krupkova97, krupkova08, morandi90, santilli78, sarlet82}.

In our approach we look for solutions of the inverse problem in
terms of semi-basic 1-forms, see Theorem \ref{LVF} and Proposition
\ref{prop:LVF}. In the Lagrangian case, the Lagrangian function
$L$ is determined by the fact that its Poincar\'e-Cartan 1-form
coincides or it is equivalent (modulo $dt$) to a semi-basic 1-form
that satisfies certain conditions. We will call these conditions
\emph{Helmholtz conditions} and we will present how they lead to
the classic formulation of Helmholtz conditions in terms of a
multiplier matrix.

\subsection{Poincar\'e-Cartan $1$-forms}

\begin{defn} \label{def:regularl}
1) A smooth function $L\in C^{\infty}(J^{1}\pi)$ is called a \emph{Lagrangian
function}.

2) The \emph{Poincar\'e-Cartan 1-form} of the Lagrangian $L$ is
the semi basic $1$-form $\theta_{L}=Ldt+d_{J}L.$

3) If for a Lagrangian $L,$ the \emph{Poincar\'e-Cartan 2-form}
$d\theta_L$ has maximal rank $2n,$ then $L$ is called a
\emph{regular Lagrangian}.
\end{defn}
For a Lagrangian function $L,$ the Poincar\'e-Cartan 2-form
$d\theta_L$ can be written as follows, see \cite{crampin84a}
\begin{eqnarray} d\theta_L=\frac{\partial^{2}L}{\partial y^{i}\partial
y^{j}}\delta y^i \wedge \delta x^j.\label{dtl} \end{eqnarray}
Therefore, the Lagrangian function is regular if and only if the
$n\times n$ symmetric matrix with local components
\begin{equation} g_{ij}(t,x,y)=\frac{\partial^{2}L}{\partial
y^{i}\partial y^{j}}\label{eq:tensorg_ij}\end{equation} has rank
$n$ on $J^{1}\pi.$ Functions $g_{ij},$ given by formula
\eqref{eq:tensorg_ij}, are the components of a $(0,2)$-type
symmetric tensor $g=g_{ij}\delta x^{i}\otimes\delta x^{j}$, which
is called the \emph{metric tensor} of the Lagrangian function $L.$

Next lemma gives characterizations of Poincar\'e-Cartan 1-forms,
as well as equivalent (modulo $dt$) Poincar\'e-Cartan 1-forms, as
subsets of semi-basic 1-forms, using the exterior differential
operator $d_J$. In the time independent case, such
characterization appears in \cite{marmo89}.
\begin{lem}
\label{LemPoincare2} Let $\theta$ be a semi-basic 1-form on
$J^{1}\pi$.

i) $\theta$ is the Poincar\'e-Cartan 1-form of a Lagrangian
function if and only if $\theta$ is $d_{J}$-closed. Moreover, the
Lagrangian function is given by $L=i_{S}\theta.$

ii) $\theta$ is equivalent (modulo $dt$) to the Poincar\'e-Cartan
1-form of a Lagrangian function if and only if $\theta$ is
$d_{J}$-closed (modulo $dt$).
\end{lem}
\begin{proof}
i) If $\theta=Ldt+d_{J}L$ is the Poincar\'e-Cartan 1-form of a
Lagrangian function $L$, then it follows that $d_{J}\theta=0$ and
$L=i_{S}\theta.$

Conversely, let us assume that $\theta\in\Lambda^{1}(J^{1}\pi)$ is
semi-basic and $d_{J}\theta=0.$ Since
$d_{J}i_{S}+i_{S}d_{J}=\mathcal{L}_{JS}-i_{[S,J]}=i_{h-S\otimes
dt-v}$, we have that $d_{J}i_{S}\theta=\theta-i_{S}\theta dt$.
Therefore $\theta=i_{S}\theta dt+d_{J}i_{S}\theta$ and hence
$\theta$ is the Poincar\'e-Cartan 1-form of the Lagrangian
$L=i_{S}\theta.$

ii) We apply the Poincar\'e-type Lemma \ref{lem:(Poincare)} for
$k=1$. Suppose that the $1$-form $\theta$ is equivalent (modulo
$dt$) to the Poincar\'e-Cartan 1-form of the Lagrangian function
$L$. Therefore $\theta\wedge dt=d_JL\wedge dt$, which means that
$\theta$ is $d_J$-exact (modulo $dt$) and hence it is $d_J$-closed
(modulo $dt$).

Conversely, if the semi-basic $1$-form $\theta$ is $d_J$-closed
(modulo $dt$), it follows that there exists a (locally defined)
Lagrangian function $L$ such that $\theta \wedge dt=d_JL\wedge
dt=\theta_L\wedge dt$. Therefore, $\theta$ is equivalent (modulo
$dt$) with the Poincar\'e-Cartan 1-form of the Lagrangian function
$L$.
\end{proof}

Next Lemma will be applied in the next section in order to
formulate Helmholtz-type conditions for the semispray $S$.
\begin{lem} \label{lemaderLie} Let $\theta$ be a semi-basic, $d_J$-closed, 1-form on
$J^{1}\pi$. Then, the next equivalent conditions are satisfied.
\begin{itemize}
\item[i)] $\mathcal{L}_{S}\theta=di_{S}\theta+i_{S}d_{h}\theta$;
\item[ii)] $i_{S}d\theta=i_{S}d_{h}\theta$;
\item[iii)] $i_{S}d_{v}\theta=0$. \end{itemize}
\end{lem}
\begin{proof}
Consider $\theta$ a semi-basic 1-form on $J^{1}\pi$ such that
$d_{J}\theta=0.$ Using Lemma \ref{LemPoincare2}, we deduce
$i_{[S,h]}\theta=i_{\mathbb{F}}\theta=i_{\mathbb{F}}(i_{S}\theta
dt+d_{J}i_{S}\theta)=i_{\mathbb{F}}d_{J}i_{S}\theta$.

Since
$i_{\mathbb{F}}d_{J}-d_{J}i_{\mathbb{F}}=d_{J\circ\mathbb{F}}-i_{[\mathbb{F},J]}=d_{v}-i_{[\mathbb{F},J]},$
it results that $i_{\mathbb{F}}d_{J}i_{S}\theta =
d_{J}i_{\mathbb{F}}i_{S}\theta + d_{v}i_{S}\theta
-i_{[\mathbb{F},J]}i_{S}\theta = d_{v}i_{S}\theta$. Hence,
$i_{[S,h]}\theta=d_{v}i_{S}\theta$. If we substitute this in
formula \eqref{eq:derLie} we obtain
\begin{eqnarray*}
\mathcal{L}_{S}\theta & = & d_{h}i_{S}\theta+i_{S}d_{h}\theta+d_{v}i_{S}\theta \\
 & = & i_{h}di_{s}\theta+i_{v}di_{s}\theta+i_{S}d_{h}\theta\\
 & = & i_{\operatorname{Id}}di_{S}\theta+i_{S}d_{h}\theta = di_{S}\theta+i_{S}d_{h}\theta,\end{eqnarray*}
which proves condition i).

Evidently conditions $i)$ and $ii)$ are equivalent due to Cartan's
formula $\mathcal{L}_{S}=d\circ i_{S}+i_{S}\circ d$.

From $ii)$ and \eqref{eq:isdht}, we obtain
$i_{\operatorname{Id}}i_{S}d\theta =
i_{S}d\theta=i_{h}i_{S}d\theta$ and hence $i_{v}i_{S}d\theta=0$.
Using now the commutation rule $i_{S}i_{v}-i_{v}i_{S}=i_{vS}=0,$
we deduce that last two conditions are equivalent. \end{proof}

\subsection{Lagrangian semisprays and dual symmetries}

For a semispray $S,$ its \emph{geodesics}, given by the system
\eqref{sode} of SODE  coincide with the solutions of the
\emph{Euler-Lagrange equations} \eqref{el_equations} of a regular
Lagrangian $L$ if and only if the two sets of equations are
related by
\begin{equation}
g_{ij}\left(t,x,\frac{dx}{dt}\right)\left(\frac{d^{2}x^{j}}{dt^{2}}
+2G^{j}\left(t,x,\frac{dx}{dt}\right)\right) =
\frac{d}{dt}\left(\frac{\partial L}{\partial y^{i}}\right) -
\frac{\partial L}{\partial x^{i}},\label{eq:SEL}\end{equation}
with $g_{ij}$ given by formula \eqref{eq:tensorg_ij}. Therefore,
for a semispray $S$, there exists a regular Lagrangian function
$L$ that satisfies equation \eqref{eq:SEL} if and only if
\begin{equation} S\left(\frac{\partial L}{\partial
y^{i}}\right)-\frac{\partial L}{\partial
x^{i}}=0.\label{eq:geod1}\end{equation} Equations \eqref{eq:geod1}
can be globally expressed as \begin{equation}
\mathcal{L}_{S}\theta_{L}=dL.\label{eq:LSTL}\end{equation} In view
of Cartan's formula, above equation \eqref{eq:LSTL} is equivalent
to $i_Sd\theta_L=0$ and hence the regularity of the Lagrangian
function $L$ from Definition \ref{def:regularl} is equivalent to
the non-degeneracy of the Poincar\'e-Cartan $1$-form $\theta_L$
from Definition \ref{def:regtheta}. For details regarding
regularity aspects of Lagrangian systems see \cite[chapter
6]{krupkova97}.
\begin{defn}
A semispray $S$ is called a \emph{Lagrangian vector field} (or a
Lagrangian semispray) if and only if there exists a (locally
defined) regular Lagrangian $L$  that satisfies equation
\eqref{eq:LSTL}.
\end{defn}

Next theorem provides necessary and sufficient conditions, in
terms of a semi-basic $1$-form, for a semispray to be a Lagrangian
vector field.  It corresponds the characterizations in terms of
$2$-forms of Lagrangian vector field in \cite{anderson92,
balachandran80, crampin81, crampin84a, marmo89}. In
\cite{sarlet95}, Proposition 8.3 gives a characterization of a
Lagrangian vector field in terms of Poincar\'e-Cartan $1$-forms.
\begin{thm}
\label{LVF} A semispray $S$ is a Lagrangian vector field if and
only if there exists a non-degenerate, semi-basic $1$-form
$\theta$ on $J^1\pi$ such that $\mathcal{L}_S\theta$ is closed.
\end{thm}
\begin{proof}
We assume that the semispray $S$ is a Lagrangian vector field for
some regular Lagrangian function $L$. Since $L$ is regular it
follows that its Poincar\'e-Cartan 1-form $\theta_L=Ldt+d_{J}L$ is
non-degenerate. Moreover, $L$ satisfies equation \eqref{eq:LSTL},
which means that $\mathcal{L}_{S}\theta_L$ is exact and hence it
is a closed $1$-form.

For the converse, consider a non-degenerate, semi-basic 1-form
$\theta$ on $J^{1}\pi,$ such that $\mathcal{L}_{S}\theta$ is
closed. It follows that there exists a (locally defined)
Lagrangian function $L$ on $J^{1}\pi$ such that
\begin{equation}
\mathcal{L}_{S}\theta=dL.\label{eq:LSDL}\end{equation} If we apply
$i_{S}$ to both sides of formula \eqref{eq:LSDL} we obtain
\begin{equation} S(i_{S}\theta)=S(L).\label{eq:SiS}\end{equation}
Now, we apply $i_{J}$ to both sides of formula \eqref{eq:LSDL} and
obtain
\begin{equation}
i_{J}\mathcal{L}_{S}\theta=d_{J}L.\label{eq:iJLST}\end{equation}
Using
$i_{J}\mathcal{L}_{S}-\mathcal{L}_{S}i_{J}=-i_{[S,J]}=i_{h-S\otimes
dt-v}$, $i_{S\otimes dt}\theta=i_{S}\theta dt$ and the fact that
$\theta$ is a semi-basic 1-form, which means that
$i_{v}\theta=i_{J}\theta=0$ and $i_{h}\theta=\theta$, from formula
\eqref{eq:iJLST} we obtain \begin{equation} \theta=i_{S}\theta
dt+d_{J}L.\label{eq:37}\end{equation} From formula \eqref{eq:37}
it follows that $d\theta=d(i_S\theta)\wedge dt + dd_JL$, which can
be written as $d\theta+i_Sd\theta \wedge dt = \mathcal{L}_S\theta
\wedge dt + dd_JL=d\theta_L$. Hence the non-degeneracy of the
semi-basic $1$-form $\theta$ implies the regularity of the
Lagrangian function $L$. We will prove now that
$\mathcal{L}_{S}\theta_{L}=dL.$ Using formula \eqref{eq:37}, we
have $\theta_{L}=\theta+(L-i_{S}\theta)dt$ and hence, according to
formulae \eqref{eq:LSDL} and \eqref{eq:SiS}, we get
\begin{eqnarray*}
\mathcal{L}_{S}\theta_{L} & = &
\mathcal{L}_{S}\theta+\mathcal{L}_{S}(L-i_{S}\theta)dt =
dL,\end{eqnarray*} which completes the proof of the theorem.
\end{proof}

For a semispray $S$ let us introduce the set
\begin{eqnarray*}
\Lambda^1_S(J^1\pi)=\{\theta \in \Lambda^1(J^1\pi), \theta
\textrm{ semi-basic, non-degenerate, } \mathcal{L}_Sd\theta=0\}.
\label{lambda1s}
\end{eqnarray*}
Theorem \ref{LVF} states that a semispray $S$ is a Lagrangian
vector field if and only if $\Lambda^1_S(J^1\pi)\neq \emptyset$.
For a Lagrangian semispray $S$, on the set $\Lambda^1_S(J^1\pi)$
we introduce the following equivalence relation $\theta_1 \equiv
\theta_2$  if $\theta_1\wedge dt = \theta_2 \wedge dt$ and
$S\left(i_S(\theta_1-\theta_2)\right)=0$. It follows immediately
that $\theta_1 \equiv \theta_2$ if and only if there exists a
first integral $f$ for $S$ such that $\theta_1=\theta_2+fdt$.

From formulae \eqref{eq:LSDL} and \eqref{eq:37} it follows that
each equivalence class $[\theta]$  in $\Lambda^1_S(J^1\pi)$
contains exactly one Poincar\'e-Cartan $1$-form $\theta_L$ of some
regular Lagrangian $L$, where
\begin{eqnarray} \theta=\theta_{L}+(i_{S}\theta-L)dt. \label{istl} \end{eqnarray}
Therefore for each equivalence class $[\theta]$  in
$\Lambda^1_S(J^1\pi)$ there exists a unique Lagrangian $L$ such
that $[\theta]=[\theta_L]$. One can furthermore reformulate this
using Lemma \ref{LemPoincare2}, each equivalence class in
$\Lambda^1_S(J^1\pi)$ contains exactly one $d_J$-closed semi-basic
$1$-form.

If an equivalence class $[\theta_L]$ contains a semi-basic
$1$-form $\theta$, which is not $d_J$-closed, then $i_S\theta-L$
is a first integral and $i_Sd\theta$ is a dual symmetry for $S$.
In this case the Lagrangian vector field $S$ is a conservative
vector field.

According to the above discussion, we can strengthen the
conclusion of Theorem \ref{LVF} as follows.
\begin{prop} \label{prop:LVF} Consider $S$ a semispray.
\begin{itemize} \item[i)] $S$ is a Lagrangian vector field if and only if there
exists $\theta \in \Lambda^1_S(J^1\pi)$ such that $d_J\theta=0$.
In this case $\theta=\theta_L$, for some locally defined
Lagrangian function $L$, and $L_S\theta=dL$.
\item[ii)] $S$ is a conservative Lagrangian vector field if and only if there
exists $\theta \in \Lambda^1_S(J^1\pi)$ such that
$d_J\theta\neq0$. In this case $\theta=\theta_L+fdt$, for some
locally defined Lagrangian function $L$ and $f$ a conservation law
for $S$.
\end{itemize}
\end{prop} Necessary and sufficient conditions for the existence
of semi-basic $1$-forms that satisfy either condition i) or
condition ii) of Proposition \ref{prop:LVF} will be discussed in
the next section in Theorems \ref{thm:LST1} and \ref{thm:LST2}.
However, we want to emphasize that a semi-basic $1$-form $\theta
\in \Lambda^1_S(J^1\pi)$ which is not $d_J$-closed gives rise to a
Lagrangian function, a first integral and a dual symmetry of the
semispray, see Theorem \ref{thm:LST2}.

\section{Helmholtz-type conditions} \label{sec_Helm}

In this section we use Fr\"olicker-Nijenhuis theory on $J^{1}\pi$
and geometric objects associated to a semispray $S$, to obtain
invariant conditions for a semi-basic 1-form $\theta$ on
$J^{1}\pi$ that are equivalent with the condition that
$\mathcal{L}_{S}\theta$ is a closed $1$-form. Therefore, in view
of Theorem \ref{LVF}, we obtain necessary and sufficient
conditions, in terms of a semi-basic $1$-form, for a semispray $S$
to be a Lagrangian vector field. We will relate these conditions
with the classic formulation of Helmholtz conditions in terms of a
multiplier matrix in the next section.

\subsection{Semi-basic $1$-forms, symmetries, and Helmholtz-type conditions}
In this section we present two theorems that give
characterizations for those semi-basic $1$-forms in the set
$\Lambda^1_S(J^1\pi)$. First theorem seeks for a solution of the
inverse problem on a restricted class of semi-basic $1$-forms:
those forms that are $d_{J}$-closed, and hence represent the
Poincar\'e-Cartan forms for some Lagrangian functions on
$J^{1}\pi$. Second theorem seeks for a solution of the inverse
problem on a larger class of semi-basic $1$-forms, which are not
$d_J$-closed. It is important to note that in this case, if there
is a solution then it induces a dual symmetry and a first integral
of the given semispray.

\begin{thm}
\label{thm:LST1} Let $\theta$ be a semi-basic, $d_J$-closed,
1-form on $J^{1}\pi$. Then, the following conditions are
equivalent
\begin{itemize}
\item[i)] $\mathcal{L}_{S}\theta$ is closed;
\item[ii)] $\mathcal{L}_{S}\theta$ is exact;
\item[iii)] $\mathcal{L}_{S}\theta=di_{S}\theta$;
\item[iv)] $d_{h}\theta=0$.
\end{itemize}
\end{thm}
\begin{proof}
Implications $iii)\Rightarrow ii)\Rightarrow i)$ are immediate.
Therefore it remains to prove implications $iv)\Rightarrow iii$)
and $i)\Rightarrow iv)$.

We prove first that condition $iv)$ implies condition $iii).$
Consider $\theta$ a semi-basic 1-form on $J^{1}\pi$ such that
$d_{J}\theta=d_{h}\theta=0.$ According to Lemma \ref{lemaderLie}
we have that $d_{J}\theta=0$ implies
$\mathcal{L}_{S}\theta=di_{S}\theta+i_{S}d_{h}\theta$. Since
$d_{h}\theta=0$ it results $\mathcal{L}_{S}\theta=di_{S}\theta$,
which is condition $iii)$.

Finally, we have to prove implication $i)\Rightarrow iv).$ If
$\mathcal{L}_{S}\theta$ is closed, we have that
$\mathcal{L}_{S}d\theta=0$ and hence
$i_{J}\mathcal{L}_{S}d\theta=0$. Using the commutation rule
$i_{J}\mathcal{L}_{S}-\mathcal{L}_{S}i_{J}= i_{\Gamma-S\otimes
dt}$ and formula $i_{J}d\theta=d_{J}\theta=0$ it results
\begin{eqnarray}
i_{\Gamma-S\otimes dt}d\theta=0. \label{igsdt} \end{eqnarray} Now,
we have $i_{\Gamma}d\theta=i_{2h-\operatorname{Id}}d\theta =
2i_{h}d\theta-i_{\operatorname{Id}}d\theta=2(i_{h}d\theta-d\theta)=2(i_{h}d\theta-di_{h}\theta)=2d_{h}\theta.$
If we substitute this in formula \eqref{igsdt} we obtain
\begin{equation} i_{S\otimes
dt}d\theta=2d_{h}\theta.\label{eq:(2)}\end{equation}

If we apply $i_{S}$ to both sides of equation \eqref{igsdt} and
use the identity $i_{S}i_{\Gamma-S\otimes dt}-i_{\Gamma-S\otimes
dt}i_{S}=0$, we obtain
\begin{equation}
i_{\Gamma-S\otimes dt}i_{S}d\theta=0.\label{eq:(1)}\end{equation}

If we apply $i_{S}$ to both sides of the equation \eqref{eq:(2)},
use condition ii) from Lemma \ref{lemaderLie}, and commutation
formula $i_{S}i_{S\otimes dt}-i_{S\otimes dt}i_{S}=i_{S}$ we
obtain $2i_{S}d_h\theta=i_{S\otimes dt}i_{S}d\theta+i_{S}d\theta$,
which implies $i_{\operatorname{Id}}(i_{S}d\theta)=i_{S\otimes
dt}i_{S}d\theta$. Last formula is equivalent to
\begin{equation}
i_{\operatorname{Id}-S\otimes dt}i_{S}d\theta=0.
\label{eq:(3)}\end{equation} Equations \eqref{eq:(1)} and
\eqref{eq:(3)} lead to \begin{equation}
i_{h}i_{S}d\theta=0.\label{eq:(4)}\end{equation} Condition iii)
from Lemma \ref{lemaderLie} can be written as \begin{equation}
i_{v}i_{S}d\theta=0.\label{eq:(5)}\end{equation} Using equations
\eqref{eq:(4)} and \eqref{eq:(5)} we obtain $i_{S}d\theta=0$.
Since $i_{S\otimes dt}d\theta=(i_{S}d\theta)\wedge dt=0,$ from
\eqref{eq:(2)} we obtain that $d_{h}\theta=0$, which is condition
iv). \end{proof}

Condition $iii)$ is equivalent with condition $i_{S}d\theta=0$
required in all previous works that deals with the inverse problem
of the calculus of variations for the time dependent case
\cite{anderson92, crampin84a, marmo89, sarlet95}. For example
condition $i_{S}d\theta=0$ is reflected in the expression of the
$2$-form $\omega$ in formula (3.2) in \cite{anderson92}. The next
theorem deals with a larger class of semi-basic 1-forms, where
condition $i_{S}d\theta=0$ is not required. In this case, in view
of Proposition \ref{prop:LVF}, we obtain a characterization for
conservative Lagrangian vector fields.

\begin{thm}
\label{thm:LST2} Let $\theta$ be a semi-basic 1-form on $J^1\pi$,
which is not $d_J$-closed. Then $\mathcal{L}_{S}\theta$ is a
closed $1$-form if and only if the following conditions hold true.
\begin{itemize}
\item[$(H_{1})$] $d_{J}\theta\wedge dt=0$ ($\theta$ is $d_{J}$-closed
modulo $dt$);
\item[$(H_{2})$] $d_{h}\theta\wedge dt=0$ ($\theta$ is $d_{h}$-closed
modulo $dt$);
\item[$(H_{3})$] $d_{\Phi}\theta\wedge dt=0$ ($\theta$ is $d_{\Phi}$-closed
modulo $dt$);
\item[$(H_{4})$] $\nabla d\theta\wedge dt=0$;
\item[$(DS)$] The 1-form $i_{S}d\theta$ is a dual symmetry for
$S.$\end{itemize}
\end{thm}
\begin{proof}
Suppose that $\mathcal{L}_{S}\theta$ is a closed $1$-form. If we
apply $i_J$ to $\mathcal{L}_Sd\theta=0$ and use commutation rule
\eqref{eq:com2} we obtain
\begin{eqnarray}
\mathcal{L}_{S}i_{J}d\theta+i_{\Gamma-S\otimes dt}d\theta=0.
\label{lsijdt}
\end{eqnarray}
If we apply again $i_J$ to above formula and use commutation rule
\eqref{eq:com2} we obtain
\begin{eqnarray*}
\mathcal{L}_{S}i_{J}d_{J}\theta+i_{\Gamma-S\otimes
dt}i_{J}d\theta+i_{J}i_{\Gamma-S\otimes dt}d\theta=0.
\end{eqnarray*}
Since $\theta$ is semi-basic it follows that
$i_Jd_J\theta=i^2_Jd\theta=2!d\theta \circ J^*=0$. Using
commutation rule \eqref{eq:com4}, $J\circ (\Gamma-S\otimes dt)=J$,
and $(\Gamma-S\otimes dt)\circ J=-J$, we get $$0 =
2i_{J}i_{\Gamma-S\otimes dt}d\theta + i_{J\circ(\Gamma-S\otimes
dt)} d\theta - i_{(\Gamma-S\otimes dt)\circ J} d\theta  =
i_{J}i_{\Gamma-S\otimes dt}d\theta+i_{J}d\theta.$$ Therefore, we
have \begin{equation}
d_{J}\theta+i_{J}i_{\Gamma}d\theta-i_{J}i_{S\otimes
dt}d\theta=0.\label{eq:DJTeta1}\end{equation} Since $
i_{\Gamma}d\theta=2d_{h}\theta$ is a semi-basic 2-form, it follows
that $i_{J}i_{\Gamma}d\theta=0$. Also, $i_{J}i_{S\otimes
dt}d\theta=i_{S\otimes dt}i_{J}d\theta=i_{S\otimes
dt}d_{J}\theta$. Formula \eqref{eq:DJTeta1} becomes
\begin{eqnarray} d_{J}\theta=i_{S\otimes
dt}d_{J}\theta.\label{eq:ih0djt}\end{eqnarray} Applying Lemma
\ref{omegadt} we obtain $d_{J}\theta\wedge dt=0$, which is
condition $(H_{1})$.

If we use formulae \eqref{eq:ih0djt}, \eqref{lsijdt} and
commutation rule $\mathcal{L}_Si_{S\otimes dt}= i_{S\otimes dt}
\mathcal{L}_S$, we obtain $i_{\Gamma}d\theta=i_{S\otimes
dt}(d\theta-\mathcal{L}_Sd_J\theta)$. Therefore,
$i_{\Gamma}d\theta\wedge dt=0$ and hence $d_h\theta\wedge dt=0$,
which is condition $(H_{2})$.

From the action of the dynamical covariant derivative $\nabla$,
expressed by formula \eqref{eq:27}, on $2$-forms we obtain
\begin{equation} \mathcal{L}_{S}d\theta=\nabla
d\theta+i_{\Psi}d\theta.\label{eq:nabladteta}\end{equation}
Applying $i_{\Gamma}$ to both sides of this identity and using
$\nabla i_{\Gamma}=i_{\Gamma}\nabla$, we obtain $ \nabla
i_{\Gamma}d\theta+i_{\Gamma}i_{\Psi}d\theta = 0$, which in view of
commutation formula \eqref{eq:com4} is equivalent to $\nabla
i_{\Gamma}d\theta+i_{\Psi}i_{\Gamma}d\theta+i_{\Psi\circ\Gamma-\Gamma\circ\Psi}d\theta=0$.
From Proposition \ref{pro:The-dynamical-covariant} we obtain
$\nabla\Gamma=0$, which implies
$\Psi\circ\Gamma-\Gamma\circ\Psi=-\mathcal{L}_{S}\Gamma=-2\mathcal{L}_{S}h
 =  -2(\mathbb{F}+J+\Phi)$ and hence
\begin{eqnarray}
\nabla i_{\Gamma}d\theta +
i_{\Psi}i_{\Gamma}d\theta-2i_{\mathbb{F}+J+\Phi}d\theta & = &
0.\label{eq:nablaigamadteta}\end{eqnarray} Using formula
\eqref{eq:nablaigamadteta} and $\nabla i_{\Gamma}d\theta\wedge
dt=i_{\Psi}i_{\Gamma}d\theta\wedge dt=0$ we obtain
$(i_{\mathbb{F}+J+\Phi}d\theta)\wedge dt=0$.

From second formula \eqref{eq:nabladteta} it results that $(\nabla
d\theta+i_{\mathbb{F}+J-\Phi}d\theta)\wedge dt=0$. Therefore
\begin{equation}
\begin{cases}
(\nabla d\theta+2i_{\mathbb{F}+J}d\theta)\wedge dt & =0,\\
(\nabla d\theta-2i_{\Phi}d\theta)\wedge dt &
=0.\end{cases}\label{eq:nabla1}\end{equation} From formula
\eqref{eq:nabla1}, it follows that there exists a 1-form $\omega$
on $J^{1}\pi$ such that $\nabla
d\theta-2i_{\Phi}d\theta=\omega\wedge dt$. Applying $i_{h}$ to
this identity, we get \begin{equation} \nabla
i_{h}d\theta-2i_{h}i_{\Phi}d\theta=i_{h}\omega\wedge
dt+\omega\wedge dt.\label{eq:nablaih}\end{equation}

We also have
$i_{h}i_{\Phi}d\theta=i_{\Phi}i_{h}d\theta+i_{\Phi\circ
h-h\circ\Phi} d\theta=i_{\Phi}i_{h}d\theta+i_{\Phi}d\theta$.

Since $i_{h}d\theta\wedge dt=d\theta\wedge dt$ $(H_{2})$, it exists
a 1-form $\tilde{\omega}$ on $J^{1}\pi$ such that $i_{h}d\theta=d\theta+\tilde{\omega}\wedge dt$.
It results that \begin{equation}
i_{h}i_{\Phi}d\theta=2i_{\Phi}d\theta+i_{\Phi}\tilde{\omega}\wedge dt.\label{eq:ihiphi}\end{equation}

Formulae \eqref{eq:nablaih} and \eqref{eq:ihiphi} lead to
\begin{equation} (\nabla d\theta-4i_{\Phi}d\theta)\wedge
dt=0.\label{eq:nabla2}\end{equation}

Relations \eqref{eq:nabla1} and \eqref{eq:nabla2} imply
$i_{\Phi}d\theta\wedge dt=0$, which is condition $(H_{3})$ as well
as $\nabla d\theta\wedge dt=0$, which is condition $(H_{4})$.

Evidently, the assumption $\mathcal{L}_{S}\theta$ closed implies
$\mathcal{L}_{S}i_{S}d\theta=i_{S}\mathcal{L}_{S}d\theta=0$, and
hence condition $(DS)$ is also satisfied.

For the converse, we assume that there exists a semi-basic 1-form
$\theta$ on $J^{1}\pi$ such that conditions $(H_{1})-(H_{4})$ and
$(DS)$ are verified. We will prove that $\mathcal{L}_{S}\theta$ is
closed.

Using condition $(DS)$ it follows that $i_S(\mathcal{L}_Sd\theta
\wedge dt)=\mathcal{L}_S(i_Sd\theta)\wedge dt +
\mathcal{L}_Sd\theta = \mathcal{L}_Sd\theta.$ Therefore, it
suffices to show that $\mathcal{L}_Sd\theta \wedge dt=0$. Using
formula \eqref{eq:27} and condition $(H_{4})$ we obtain
$$ \mathcal{L}_Sd\theta \wedge dt=(\nabla d\theta
+i_{\Psi}d\theta)\wedge dt = i_{\Psi}d\theta \wedge dt.$$ We will
show that the $2$-form $i_{\Psi}d\theta \wedge dt$ vanishes by
evaluating it on pairs of strong horizontal and/or vertical vector
fields.

From conditions $(H_{1})-(H_{3})$ we obtain that the $2$-forms
$d_J\theta$, $d_h\theta$ and $d_{\Phi}\theta$ vanish on any pair
of strong horizontal vector fields $h_1X, h_1Y$. From condition
$(H_{1})$ we have $i_Jd\theta(h_1X, h_1Y)=0$ and hence
\begin{eqnarray}
d\theta(JX, h_1Y)+ d\theta(h_1X, JY)=0. \label{dtjh}
\end{eqnarray}
From condition $(H_{2})$ it follows that $(i_hd\theta -
d\theta)(h_1X, h_1Y)=0$, which implies
\begin{eqnarray}
d\theta(h_1X, h_1Y)=0. \label{dthh}
\end{eqnarray}
From condition $(H_{3})$ we obtain that $i_{\Phi}d\theta(h_1X,
h_1Y)=0$, which implies
\begin{eqnarray}
d\theta(\Phi X, h_1Y)+ d\theta(h_1X, \Phi Y)=0. \label{dtph}
\end{eqnarray}

Using formulae \eqref{dtjh}, \eqref{dthh} and \eqref{dtph} we
obtain
\begin{eqnarray*}
(i_{\Psi}d\theta)(JX, JY) & = & d\theta(JX, h_1Y)+ d\theta(h_1X,
JY)=0,
\nonumber \\
(i_{\Psi}d\theta)(h_1X, h_1Y) & = & d\theta(-\Phi X, h_1Y)+
d\theta(h_1X, -\Phi Y)=0, \label{ipdtjj} \\
(i_{\Psi}d\theta)(JX, h_1Y) & = & d\theta(h_1X, h_1Y)+ d\theta(JX,
-\Phi Y)=0, \nonumber
\end{eqnarray*}
for any arbitrary pair of vector fields $X, Y$ on $J^1\pi$. Last
three formulae imply that $i_{\Psi} d\theta \wedge dt=0$ and hence
$\mathcal{L}_Sd\theta=0$.
\end{proof}
In view of Proposition \ref{prop:LVF}, one can reformulate Theorem
\ref{thm:LST2} as follows. A semispray $S$ is a Lagrangian vector
field if and only there exists a semi-basic $1$-form $\theta$ that
satisfies the Helmholtz conditions $(H_{1})-(H_{4})$. Note that in
this case we might have $d_J\theta=0$ and hence $\theta=\theta_L$
for some Lagrangian function $L$. If $d_J\theta\neq 0$ then
$\theta$ is equivalent (modulo $dt$) with the Poincar\'e-Cartan
$1$-form of some Lagrangian function. In this case, $S$ is a
conservative Lagrangian vector field and $i_Sd\theta$ is a dual
symmetry.

\subsection{Semi-basic 1-forms and multiplier matrices}
In this section we present a proof of the Theorem \ref{thm:LST2}
using local coordinates. First, this allows to show the usefulness
of the covariant derivative studied in Section \ref{dcv}, as well
as the use of the adapted basis and cobasis \eqref{eq:11}.
Secondly, it will relate the Helmholtz-type conditions
$(H_{1})-(H_{4})$ presented in Theorem \ref{thm:LST2} with their
classic formulation for a multiplier matrix.

Consider that $\theta=\theta_{0}dt+\theta_{i}\delta x^{i}$ is a
semi-basic 1-form on $J^{1}\pi$ that is not $d_J$-closed and such
that the $1$-form $\mathcal{L}_{S}\theta$ is closed. For the
$1$-form $\theta$ we will use the following
notations.\begin{eqnarray}
a_{i}=\frac{\partial\theta_{0}}{\partial y^{i}}-\theta_{i}, &  & b_{i}=\frac{\delta\theta_{0}}{\delta x^{i}}-\nabla\theta_{i},\label{eq:40}\\
b_{ij}=\frac{\delta\theta_{i}}{\delta
x^{j}}-\frac{\delta\theta_{j}}{\delta x^{i}}, &  &
g_{ij}=\frac{\partial\theta_{i}}{\partial y^{j}}.\nonumber
\end{eqnarray} A direct calculus using adapted cobasis \eqref{eq:11} leads to \begin{eqnarray} d\theta
& = & b_{i}\delta x^{i}\wedge dt+a_{i}\delta y^{i}\wedge
dt+\frac{1}{2}b_{ij}\delta x^{j}\wedge\delta x^{i}+g_{ij}\delta
y^{j}\wedge\delta x^{i}.\label{eq:41}\end{eqnarray} We emphasize
the presence of the two terms $a_i$ and $b_i$ in formula
\eqref{eq:41} due to the fact that $i_Sd\theta \neq 0$, terms
which do not appear in previous work for the time dependent case
of the calculus of variations, see \cite{anderson92, crampin84a,
sarlet95}.

Using formula \eqref{eq:27} for the dynamical covariant
derivative, the $2$-form $\mathcal{L}_Sd\theta=\nabla d\theta
+i_{\Psi}d\theta$ can be expressed in terms of the adapted cobasis
\eqref{eq:11} as follows.
\begin{eqnarray}
\mathcal{L}_{S}d\theta & = & (\nabla b_{i}-a_{j}R_{i}^{j})\delta x^{i}\wedge dt+(b_{i}+\nabla a_{i})\delta y^{i}\wedge dt\nonumber \\
 &  & +\frac{1}{2}(\nabla b_{ij}-g_{ik}R_{j}^{k}+g_{jk}R_{i}^{k})\delta x^{j}\wedge\delta x^{i}\label{eq:42}\\
 &  & +(\nabla g_{ij}+b_{ij}-b_{ji})\delta y^{j}\wedge\delta x^{i}+\frac{1}{2}(g_{ij}-g_{ji})\delta y^{j}\wedge\delta y^{i}.\nonumber \end{eqnarray}
From above formula \eqref{eq:42} it follows that the condition
$\mathcal{L}_{S}\theta$ is closed is equivalent with the following
two sets of conditions that correspond to Helmholtz conditions
($H_1$)-($H_4$) in Theorem \ref{thm:LST2}
\begin{eqnarray}
g_{ij} & = & g_{ji},\label{eq:43}\\
g_{ik}R_{j}^{k} & = & g_{jk}R_{i}^{k},\label{eq:44}\\
\nabla g_{ij} & = & 0,\label{eq:45}\\
b_{ij} & = & 0,\label{eq:46}
\end{eqnarray}
and respectively to condition (DS) in Theorem \ref{thm:LST2}
\begin{eqnarray}
b_{i}+\nabla a_{i} & = & 0,\label{eq:47}\\
\nabla b_{i}-a_{j}R_{i}^{j} & = & 0.\label{eq:48}\end{eqnarray} If
the above two equations \eqref{eq:46} and \eqref{eq:47} hold good
we deduce
\begin{equation}
\nabla^{2}a_{i}+a_{j}R_{i}^{j}=0\label{eq:49}\end{equation} which
is equivalent with the fact that the 1-form
\[i_{S}d\theta=\nabla a_{i}\delta x^{i}-a_{i}\delta
y^{i}=-b_{i}\delta x^{i}-a_{i}\delta y^{i}\] is a dual symmetry of
the semispray $S$, since it satisfies equation
\eqref{eq:dualsymlocal}.

Next, we present the local expressions of the $2$-forms
$d_J\theta$, $d_h\theta$, $d_{\Phi}\theta$ and $\nabla d\theta$.
\begin{eqnarray*}
d_{J}\theta & = & a_{i}\delta x^{i}\wedge dt + \frac{1}{2}(g_{ij}-g_{ji})\delta x^{j}\wedge\delta x^{i},\nonumber \\
d_{h}\theta & = & b_{i}\delta x^{i}\wedge dt+\frac{1}{2}b_{ij}\delta x^{j}\wedge\delta x^{i},\nonumber \\
d_{\Phi}\theta & = & R_{i}^{j}a_{j}\delta x^{i}\wedge dt+\frac{1}{2}(g_{ik}R_{j}^{k}-g_{jk}R_{i}^{k})\delta x^{j}\wedge\delta x^{i},\label{eq:der}\\
\nabla d\theta & = & \nabla b_{i}\delta x^{i}\wedge dt+\nabla
a_{i}\delta y^{i}\wedge dt+\frac{1}{2}\nabla b_{ij}\delta
x^{j}\wedge\delta x^{i} +\nabla g_{ij}\delta y^{j}\wedge\delta
x^{i}.\nonumber \end{eqnarray*} In view of above formulae, one can
immediately see that Helmholtz-type conditions $(H_{1})-(H_{4})$
are equivalent with conditions \eqref{eq:43}-\eqref{eq:46}. First
three conditions \eqref{eq:43}-\eqref{eq:45} are usually known as
the classic Helmholtz conditions of the multiplier matrix
$g_{ij}=\partial \theta_i/\partial y^j$. Fourth classic Helmholtz
condition
$$ \frac{\partial g_{ij}}{\partial y^{k}}= \frac{\partial g_{ik}}{\partial
y^{j}}$$ is identically satisfied in view of last notation
\eqref{eq:40}.

The requirement that semi-basic $1$-form $\theta$ is not
$d_J$-closed implies that $i_Sd_J\theta\neq 0$. Therefore
$i_Ji_Sd\theta\neq 0$ and hence $i_Sd\theta \neq 0$, which assures
that a solution of equation \eqref{eq:49} is not trivial. Locally,
$d_J\theta \neq 0$ implies  $a_i$ from formula \eqref{eq:40} is
not identically zero, and hence $i_{S}d\theta=\nabla a_{i}\delta
x^{i}-a_{i}\delta y^{i}$ is a non trivial, dual symmetry of the
semispray $S$.

Theorem \ref{thm:LST1} corresponds to the time independent case
studied in Theorem 4.3, \cite{bucataru09}, where it is shown that
for $0$-homogeneous semi-basic $1$-forms only two of the Helmholtz
conditions are independent. These two conditions are $d_J\theta=0$
and $d_h\theta=0$ and appear in both Theorem \ref{thm:LST1} as
well as Theorem 4.3 from \cite{bucataru09}.

Theorem \ref{thm:LST2} corresponds to the time independent case
studied in Theorem 4.1, \cite{bucataru09}, where conditions
$(H_{1})-(H_{4})$ are equivalent with the fact that a semi-basic
$1$-form is $d_J$, $d_h$ and $d_{\Phi}$-closed and satisfies
$\nabla d\theta=0$. It is important to emphasize that in general,
for the time dependent case, Lagrangian semisprays are not
conservative. Theorem \ref{thm:LST2} refers to the class of
Lagrangian semisprays (conditions $(H_{1})-(H_{4})$ are satisfied)
that have symmetries as well (condition $(DS)$ is satisfied).

\begin{acknowledgement*}
The authors acknowledge fruitful comments from O.~Krupkov\'a
regarding some regularity aspects of the inverse problem and want
to express their thanks to W. Sarlet for many discussions that
clarified the main results of the paper.

The authors have been supported by CNCSIS grant IDEI 398 from the
Romanian Ministry of Education.
\end{acknowledgement*}

redpajama_set_name: RedPajamaArXiv


text:
Le TANK 300 est un modèle de SUV compact produit par Great Wall Motors sous la marque TANK, autrefois sous WEY.

Aperçu
Le TANK 300 a été présenté en juillet 2020 au Salon de l'auto de Chengdu. Il a été mis en vente sur le marché chinois le 17 décembre 2020. C'est le premier véhicule à être construit sur la plate-forme appelée "Tank" par Great Wall.

Le TANK 300 est propulsé par un moteur essence turbo de deux litres connu des autres modèles de Wey. Il a une puissance de 167 kW (227 ch) et accélère le véhicule tout-terrain à 100 km/h en dix secondes. Wey spécifie la vitesse maximale à 170 km/h. Le véhicule dispose de trois blocages de différentiel.

Une édition Frontier est lancée en Chine en décembre 2022, avec un kit carrosserie spécifique accentuant son ambiance tout-terrain.

Exportations
Le 6 juillet 2022, le TANK 300 version tout-terrain a été lancé en Arabie saoudite.

Références

Article connexe
 Tank 500

Tank 300
Automobile des années 2020
SUV

redpajama_set_name: RedPajamaWikipedia


(Ottawa, May 11, 2017) The Justice for Abdirahman Coalition is expressing disappointment concerning the second delay in the judicial pre-trial of Constable Daniel Montsion, the Ottawa police officer charged in the 2016 death of Abdirahman Abdi. Although the pre-trial was set to begin on May 1, it was was adjourned to give the defence more time to review evidence. Yesterday's decision to approve yet another request to delay proceedings has been met with deep disappointment, both on the part of the Coalition as well as concerned members of the community at large.
Despite recognizing that such developments are commonplace in judicial proceedings, it should be noted that this case carries with it far reaching implications regarding police-community relations.
With a new pre-trial set for July 12, the Abdi family approaches the first anniversary of Abdirahman's death only to be told once again that justice will be delayed. Due process in a timely manner is all they ask.
The Coalition has been at the forefront in the calls for transparency, accountability, and justice in this case since Abdirahman's death. The Coalition strongly urges all relevant parties to undertake a concerted effort in both effectively and efficiently adjudicating this matter in such a way that would end the suffering of a family that has been too long without peace.
Communities across the province have been traumatized by this tragedy. Many are watching this case closely and hoping for restored faith in our justice system.


26 November, Abuja – On 26 and 27 November 2015, Commissioners Mary Chinery-Hesse (Ghana) and Adeolu Ogunrombi (Nigeria) will share key findings of the Commission's report Not Just in Transit: Drugs, the State and Society in West Africa with the Minister of Health, the Inter-Ministerial Committee on Drugs Control and civil society groups.
The visit of the Commissioners follows President Muhammadu Buhari acceptance speech remarks that Nigeria aims to become "a more forceful and constructive player" on global issues, such as countering the global drugs problem.
"The West Africa Commission on Drugs is ready to support Nigeria in this endeavour" says Mary Chinery-Hesse." "Drug trafficking, consumption and production in Nigeria and beyond can undermine institutions, threaten public health and damage development efforts".
"In line with the findings of our report, we call on Nigeria and the other West African governments to reform drug laws and policies and decriminalise low-level and non-violent drug offences", says Adeolu Ogunrombi.
"The glaring absence of treatment facilities for drug users fuels the spread of disease and exposes an entire generation, users and non-users alike, to growing public health risks.The current approaches have not delivered solutions and Nigeria too needs to rethink its drug policies," says Obasanjo.


text:
No. 788 - Deloraine - 'The Church of the Latter Day Saints'
Deloraine is a historic tourist town situated on the Meander River and lies approximately halfway between the cities of Launceston and Devonport. The settlement dates to the 1830s and was named by the surveyor, Thomas Scott, after Sir William Deloraine, a character in Sir Walter Scott's poem "The Lay of the Last Minstrel". Sir Walter Scott was a relative of Thomas Scott.
The Church of Jesus Christ of Latter-day Saints (LDS) dates back to 1854, when Robert Owen began to proselytise in Hobart. This mission was however unsuccessful and it was not until the 1890's that congregations was established in Hobart and Upper Huon. Chapels were built in North Hobart in 1924 and in Glen Huon in 1927. Many early converts migrated to Utah which was a cause of opposition to the 'Saints' proselytising by the Australian government.
The Latter Day Saints established a ward (parish) at Deloraine in 1995. Prior to this members of the LDS had to travel to Launceston for meetings. On Saturday 11 October 1997, construction began on a new church built in 'Casey's paddock', at the corner of Emu Bay Road and Connector Road. The first sods of earth were turned by Councillor Greg Hall, Mayor of the Meander Valley Council, and Henry and Sheila Triffitt. The building was completed and officially opened in 1898. It is the seventh LDS chapel to be built in Tasmania.
Source: Western Tiers
Western Tiers, Tuesday 18 November 1997, page 7
Western Tiers, Tuesday 28 April 1998, page 31
https://www.utas.edu.au/library/companion_to_tasmanian_history/C/Church%20of%20Jesus%20Christ..htm
1990-1999 Latter-Day Saints Northern Tasmania

redpajama_set_name: RedPajamaCommonCrawl


text:
Congress-Bundestag Youth Exchange (CBYX) Program
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Diversity in Prague
CIEE wants all our students to feel welcomed, supported, and empowered to succeed while studying abroad. On this page, local CIEE staff have provided details about conditions and cultural attitudes that students with specific identities might encounter at their location.
The information below is just a broad overview so if you have specific questions or concerns not covered here, please email inclusion@ciee.org. We would be glad to have local staff share their perspectives, talk with you about accommodations, connect you with resources, and/or put you in touch with a program alum who could speak about their experiences navigating a program in this location.
No matter where you choose to study abroad with CIEE, our staff—all of whom receive regular and comprehensive training in diversity, equity, and inclusion—will be on hand throughout your program to provide advice, resources, and support regarding these issues.
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People in the Czech Republic generally dress more moderately and conservatively than people in the United States and western Europe. However, we see Czech youth, primarily influenced by social networks and current trends being more courageous both in dressing and in the certainty of presenting their own body.
Although the Czech Republic and the city of Prague have been working on accessibility improvements for physically disabled people, cobblestones and high curbs present mobility issues, and many older buildings are still not wheelchair accessible. Prague and other large cities have been making progress on accessibility of public transport. Many trams are "low riders," and busses have wheelchair lifts or platforms. Most of Prague's underground stations have been equipped with elevators, although some in the historical city center lack elevators due to the city conservation regulations.
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Much like other countries of the region, Czech Republic is a male-dominated society. Sex-based discrimination in the workplace is illegal. However, women's employment rate is lower as are their wages. (The figures for women's employment are comparable to the U.S. data.) Large cities, such as Prague, have seen an improvement though. Women are more independent, though they are still expected to keep their job as well as manage the care of children and the household. Women might also be subject to catcalling in public spaces and unwanted advances, especially at bars and parties.
Trans and non-cis activism has recently been on the rise in the Czech Republic, mainly among younger people. Individual activists and organizations are striving for the empowerment, social justice, promotion of rights, and positive social changes for the benefit of trans men, women, and other non-cis persons; and they work towards understanding and acceptance of trans and non-cis persons on both the social and legislative level. However, the situation is far from ideal and people outside the traditional gender binary (in terms of identity and expression) remain a curiosity for many Czechs and might encounter a whole array of different reactions on a daily basis: from acceptance to indifference to rejection.
During orientation and throughout your program, CIEE staff are on hand to provide advice and support regarding issues of gender and gender identity.
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The Czech lands saw significant out-migration during the late 19th-century as the U.S. became a center of world agricultural and industrial production, and during periods of war and societal upheaval such as WWI, WWII, and following the Warsaw Pact invasion of Czechoslovakia in 1968. Many American citizens with Czechoslovak heritage have relatives who first came to the United States during one of those periods, immigrating to cities like Chicago, New York, and Pittsburgh, and to agricultural areas in states such as Texas, Iowa, Nebraska, Wisconsin, and Minnesota.
At present, a good number of Czechs study, work, and sometimes settle abroad, driven by opportunity rather than crisis. It is not uncommon to meet locals who have friends and family abroad. Taking a Czech language course is a good way for heritage-seekers to begin to connect with the region, and taking courses about the history, culture, and politics of the region can help deepen your understanding of the context of out-migration. Taking part in the excursions and cultural activities organized by local staff can also help give heritage seekers insight into the region and its traditions.
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Czech society is undergoing tremendous change. As a result of the actual world political situation, some social problems are emerging. These problems may sound familiar to the western world but have been either long suppressed in Central/East Europe or are entirely new. Some may show resentment particularly toward those who appear to be Muslims, Roma, Asian or African.
The Czech Republic is ethnically homogenous. While it is quite rare to encounter people of different racial backgrounds in rural areas, the urban centers, especially Prague are growing more and more racially diverse. However, compared to large metropolitan areas of the United States and Western Europe, the population of Prague is still largely white. The main non-white minorities are the Roma and the Vietnamese. This might cause some feelings of anxiety, especially combined with occasional staring of by-passers. However, this should be interpreted as a mere curiosity rather than a sign of aggressive behavior.
To address both overt and latent racism and discrimination – although rare – CIEE Prague staff are available to provide support and resources to help all CIEE students enjoy and learn from their study abroad experience, irrespective of – indeed, celebrating – our diverse racial and ethnic identities. CIEE Prague is sensitive to the range of these issues and will provide information and support to proactively create an inclusive community.
Although the Czech Republic boasts a large number of beautiful churches in many architectural styles, it has one of the most secular societies in the world. The churches attest to its long Christian tradition, which started in the 10th century and was marked by the Hussite defense against Catholic forces, followed by a forced re-Catholicization in the 17th century. Alongside Christians, the Jewish population was native to the Czech lands, gaining equality in 1867 and being gradually integrated into society until the Nazi genocide during World War II. The forty years of communism put an end to religious practice as the regime persecuted anyone who openly professed.
After the fall of communism, freedom of religion was granted, and everybody now has the right to express their own religion or beliefs. There is no official religion. The percentage of population belonging to a denomination is as low as 14% of all inhabitants, with another 7% declaring themselves as believers who do not belong to a denomination.
Overall, Czechs are tolerant of the religious beliefs of others although the refugee crisis in the mid-2010s resulted in expressions of Islamophobia and suspicion towards people wearing traditional clothing (hijabs etc.). During orientation and throughout your program, CIEE staff are on hand to provide advice and support regarding these issues.
Czech Republic is regularly branded as the most progressive country in the former Eastern Bloc in terms of LGBTQ+ rights. According to the latest report issued by the Czech Public Defender of Rights and the Prague Pride Association (which was based on one of the largest surveys among lesbians, gays, bisexuals and trans people in the Czech Republic), the Czech LGBTQ+ community tends to recognize that their position in the Czech Republic is rather satisfactory, however not without any issues as they still sometimes experience odd behavioral encounters (such as people staring at them in public) or hear opinions that they should not be publicly displaying their sexual orientation. Prague, however, is often considered to be a safe haven for the LGBTQ+ community in the former Eastern Bloc region. Over the past few years, the Prague LGBTQ+ community has expanded and changed significantly: from pride parades taking place every summer, to queer parties, to organizations that fight for trans rights and marriage equality.
Prague is a very affordable city compared to most Western European capitals, and its student-budget-friendly environment has long been one of its attractions for foreign students. It can be very cheap to eat out if you eat like a local, at local restaurants; it can quickly become more expensive if you go to familiar American chain restaurants in Prague. Shopping for groceries and cooking in your housing is one of the easiest ways to save money, but here too it helps to buy local goods – it's likely that familiar American brands are more expensive than what you may pay back home. Getting around town is cheap – a transportation pass, good for all public transportation in Prague (metro, tram, bus, ferry boat, funicular) is included as part of semester, summer, and January CIEE study programs. And some of the most fun things to do in the city are free: hike the national trail network in the nature preserves in and around Prague, explore the many local parks, and attend local outdoor festivals and performances. The Czech Republic has long been one of the most egalitarian among OECD countries, at least in terms of income inequality, so you don't see as much "conspicuous consumption" as you may elsewhere, though the liberalized economy of post-1989 has changed that a bit, especially in Prague.
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GogiPet ® purple dog collar with black striped dog tie size L. Luxuriously can also be your big dog on the dogwalk with the black dog tie and purple collar on the pink dog collar. GogiPet ® Dog collar with 38 cm - 45 cm neck extent.-25% offer.
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Sold in a set of two, the Cathy's Concepts Monogram Moscow Mule Copper Mugs A-Z are fashioned with varying metal tones and plated in copper. These are perfect for cocktail parties or just a breezy summer night on the patio. Made from stainless steel with a sleek copper finish, these mugs will bring a subtle rustic accent to your decor.
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Enjoy a tall glass of something cool from the Cathy's Concepts Personalized Monogram Craft Beer Mug Set. Conventionally used for a cold pilsner brew, this set of 4 pours a sip of classic personal style into your gatherings. Crafted from glass into a traditional styled mug, it features a thick handle, a short stem, and a uniquely cut design on the foot of the glass.
Add convenience and contemporary style to your kitchen with these simple and elegant Glass Stackable Mugs from Made By Design™. This set of white mugs includes six glass mugs that stack on top of each other for space-saving storage. Each mug holds up to 12.5 ounces of hot or cold liquid — whether you're enjoying coffee, hot cocoa or just plain water, your hands will stay comfortable and your kitchen will stay classy. These mugs are dishwasher and microwave-safe, making it easy to reuse them over and over again.
Complete your drinkware collection with these Mason Jar Mugs from Lillian Rose. Constructed from glass with a classic look, this pair of 16-ounce mason jar mugs with handles are perfect to hold and sip your favorite cold beverages from iced tea to fruity cocktails. This pair of glass jar mugs look elegant while their vintage charm adds to any table's aesthetics.
You can now bring home the bistro and cafe experience by serving up yummy beverages with the Luminarc 13oz Nordic Glass Mug Set of 4. With this set of four, you can experiment away with the presentation of your favorite beverage. The compact and sleek design make it a perfect accent for almost any countertop! It also comes with a cute handle making it easy to move around! It's perfect for contemporary and rustic setting alike. Made of glass, this chic set of four is easy to clean and care for and is a lovely set to serve up hot beverages like coffee or tea. Whether it quite time for yourself or entertaining family and friends, this set will always accent your beverage and the mood of the day.
Keep your drink as warm or cold for as long as it takes to enjoy it with this clear insulated Tervis Jewel Mug. Adding texture, subtle ridges run horizontally around the inside of the mug. This 16oz mug is perfect for when a quick warm-up is needed. This mug can be popped into the microwave for heating and is also dishwasher-safe for easy cleanup.
Merlot to make you go whoah! Libbey Vineyard Reserve Stemware 4pc Merlot Glasses is the perfect add on to your wine spread. Enjoy your favorite bottle of red with this elegant glass; enjoyed best with the right glass at the right temperature and setting. Don't short yourself on the pour!


This building at the southeast corner of Logan Street and Ionia Avenue SW is a former furniture factory that may be renovated for apartments by an Ann Arbor-based firm.
GRAND RAPIDS -- A century-old furniture factory south of downtown could be in for a $29 million makeover into low- and middle-income apartments.
Ann Arbor-based HDC Companies, led by East Grand Rapids native Bob Jacobson, is working on a plan that would bring 90 one- and two-bedroom apartments to the five-story brick building at 60 Logan St. SW.
Renovating the former Century and Baker Furniture plant hinges on securing historic tax credits along with funding from the Michigan State Housing Development Authority, he said.
"I'm quite sure we can put together what we need to do this," Jacobson said. "The investment community has been very positive on this."
HDC plans to redevelop the building, to be called Century Lofts, in two phases.
About 45 apartments would be built on the eastern side of the 125,000-square-foot building for the first phase.
The second phase would bring 15,000 square feet of commercial space on the ground floor, plus the balance of the apartments.
HDC hopes to close on a deal to buy the property by year's end if the MSHDA funding and tax credits come through.
The first units would be available for occupancy 15 to 16 months later.
If the project gets off the ground, it would mark the first major non-industrial investment south of Wealthy Street along the Ionia Avenue corridor.
The building is currently used for storage. Its neighbors include an automotive coating factory, a paint supply store and warehouse, the former Klingman's Furniture warehouse and several empty warehouses purchased by the Downtown Development Authority.
"It's interesting.Wealthy is where everything stopped on the southern side with the development," Jacobson said. "It's an imaginary line that became a barrier for no purpose."
Grand Rapids Economic Development Director Kara Wood said the project would be a positive addition to the area.
"I think (residential development south of Wealthy) is reasonable to expect and it's a decent location," she said. "It also would be nice to just clean that property up and breathe some new life into that building. It's a really sound structure for the most part."
Rents would be based on income, and start at about $465 a month.