The 1-dimensional Hamiltonian of a classical particle of mass m is
\(H=\frac{p^2}{2 m} e^{-x / a}+V(x) \)
where a is a constant with appropriate dimensions. The corresponding Lagrangian is,
1
\(\frac{m}{2}\left(\frac{d x}{d t}\right)^2 e^{x / a}-V(x) \)
2
\(\frac{m}{2}\left(\frac{d x}{d t}\right)^2 e^{-x / a}-V(x) \)
3
\(\frac{3 m}{2}\left(\frac{d x}{d t}\right)^2 e^{x / a}-V(x) \)
4
\(\frac{3 m}{2}\left(\frac{d x}{d t}\right)^2 e^{-x / a}-V(x) \)