The Jacobi variational principle revisited. (English) Zbl 1082.70008

Grabowski, Janusz (ed.) et al., Classical and quantum integrability. Dedicated to Włodzimierz Tulczyjew. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Cent. Publ. 59, 99-111 (2003).
Jacobi in the 19th century showed that with the help of energy conservation law the action integral of a mechanical system could be reduced to another action which is described only as the spatial path of dynamical system. The Lagrangian of this action is called Jacobi Lagrangian. Here it is shown that for the dynamical systems whose Lagrangian does not depend on time, the usual Hamiltonian system can be brought to a form which enables one to employ the Routh procedure and to eliminate from the action integral the information about temporal evolution of the system. Also the following (inverse Jacobi problem) problem is solved: given any Lagrange function \(L\) homogeneous of degree one in velocities and a function \(G\) of position coordinates and velocities, how one can find another Lagrange function \(LH\) such that \(G\) is energy function and \(L\) is Jacobi Lagrangian for \(LH\).
For the entire collection see [Zbl 1011.00047].


70H30 Other variational principles in mechanics
70H25 Hamilton’s principle
49S05 Variational principles of physics