Higher Order Hamiltonian Systems with Generalized Legendre Transformation

Abstract

The aim of this paper is to report some recent results regarding second order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. The associated 3rd 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.

On controlled Hamilton and Hamilton–Jacobi differential equations of higher-order

“…In Krupková and Smetanová 19 studied the Legendre transformation for regularizable Lagrangians in field theory. Later, Smetanová 20 stated some results regarding second-order Lagrangians corresponding to 2nd and 3rd order Euler–Lagrange forms. Also, the associated 3rd order Hamiltonian systems have been established.…”