| Abstract: |
| Given a regular contact Lagrangian $L:TQ\times \mathbb{R} \to \mathbb{R}$, we can compute its Herglotz vector field $\xi_L$, whose integral curves are lifts of the curves on $Q$ that satisfy the Herglotz variational principle. This is contact Lagrangian mechanics analog of the Euler-Lagrange vector field. We will study the following two related problems:
\begin{itemize}
\item \textbf{The problem of Equivalent Lagrangians}: given two contact Lagrangians $L,\bar{L}:TQ\times \mathbb{R} \to \mathbb{R}$, determine when is $\xi_L$ \emph{equivalent} to $\xi_{\bar{L}}$.
\item \textbf{The inverse problem}: given a vector field $X$ on $TQ\times \mathbb{R}$, find the conditions for the existence of a Lagrangian $L:TQ\times \mathbb{R} \to \mathbb{R}$ such that $X$ is \emph{equivalent} to $\xi_L$.
\end{itemize}
Obviously, the solution of this problems depends on what we mean by \emph{equivalent}. In the symplectic case, equality of the vector fields is required. However, this condition is too strong in our situation. We can use the flexibility provided by the extra $\mathbb{R}$ coordinate to define weaker notions of equivalence.
We will examine the equivalent Lagrangians problem and the inverse problem under some of this weaker notions of equivalence and provide geometric and variational characterizations of the solution. |
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