Alternative Lagrangians obtained by scalar deformations

We study non-conservative like SODEs admitting explicit Lagrangian descriptions. Such systems are equivalent to the system of Lagrange equations of some Lagrangian $L$, including a covariant force field which represents non-conservative forces. We find necessary and sufficient conditions for the existence of a differentiable function $Φ:\mathbb{R}\rightarrow\mathbb{R}$ such that the initial system is equivalent to the system of Euler-Lagrange equations of the deformed Lagrangian $Φ(L)$. We give various examples of such deformations.

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