Geometrical structures of higher-order dynamical systems and field theories

TL;DR

This thesis develops geometric formulations for higher-order dynamical systems and field theories, providing a unified framework to analyze complex physical systems using Lagrangian and Hamiltonian formalisms.

Contribution

It introduces geometric frameworks for higher-order systems and field theories, applying them to various physical examples and problems.

Findings

Unified geometric approach for higher-order systems

Application to Hamilton-Jacobi theory and relativistic particles

Analysis of Korteweg-de Vries and other field equations

Abstract

In this Thesis we develop the geometric formulations for higher-order autonomous and non-autonomous dynamical systems, and second-order field theories. In all cases, the physical information of the system is given in terms of a Lagrangian function/density, or a Hamiltonian that admits Lagrangian counterpart. These geometric frameworks are used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory.