On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.