Single-Time and Multi-Time Hamilton–Jacobi Theory Based on Higher Order Lagrangians

This paper aims to present aspects of Hamilton–Jacobi theory via single-time and multi-time higher order Lagrangians, more precisely: Hamilton–Jacobi PDE, Hamilton–Jacobi system of PDEs, Hamilton–Jacobi divergence PDE, generating function, canonical momenta, change of variables in Hamiltonian and gauge transformation. The present work can be seen as a natural continuation of a recent paper (see [12]), where only multi-time Hamilton–Jacobi theory via second-order Lagrangians is considered. Over time, many researchers have been interested in the study of Hamilton–Jacobi equations. It is well known that the classical (single-time) Hamilton–Jacobi theory appeared in mechanics or in information theory from the desire to describe simultaneously the motion of a particle by a wave and the information dynamics by a wave carrying information. Thus, the Euler–Lagrange ODEs or the associated Hamilton ODEs are replaced by PDEs that characterize the generating function. Later, using the geometric setting of the k-osculator bundle (see [5, 8]), R. Miron and M. R. Roman studied the geometry of higher order Lagrange spaces, providing some applications in mechanics and physics. Also, O. Krupkova has investigated the Hamiltonian field theory in terms of differential geometry and local coordinate formulas (see [3]). The multi-time version of Hamilton–Jacobi theory has been extensively studied by many researchers in the last few years (see [1, 6, 7, 12, 15, 18]). In this paper, we develop our points of view (see the multi-time Lagrange–Hamilton–Jacobi theory—via first-order Lagrangians—formulated and studied by C. Udrişte and his collaborators, [18]), by developing the new concepts and methods (see, for instance, Hamilton–Jacobi divergence PDE) for a theory that involves single-time and multi-time higher order Lagrangians. This work can be used as a source for research problems and it should be of interest to engineers and applied mathematicians. For more contributions and various approaches about different aspects of Lagrange-Hamilton dynamics and Hamilton-Jacobi theory, the reader is directed to [2, 4, 10, 11, 13, 14, 16, 17].