A generalization of the Liouville–Arnol'd theorem Academic Article uri icon


  • AbstractWe show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.


publication date

  • March 1995