Higher order Hamiltonian systems with generalized Legendre transformation (Q1634508)

Summary: The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to second and third order Euler-Lagrange forms. The associated third order Hamiltonian systems are found. The generalized Legendre transformation and geometrical correspondence between solutions of the Hamilton equations and the Euler-Lagrange equations are studied. The theory is illustrated on examples of Hamiltonian systems satisfying the following conditions: (a) the Hamiltonian system is strongly regular and the Legendre transformation exists; (b) the Hamiltonian system is strongly regular and the Legendre transformation does not exist; (c) the Legendre transformation exists and the Hamiltonian system is not regular but satisfies a weaker condition.