Lecture Notes in Integral Invariants and Hamiltonian Systems

TL;DR

This paper reviews the fundamental concepts of integral invariants theory, highlighting its historical development and connections across various fields like Hamiltonian dynamics, optics, and hydrodynamics.

Contribution

It provides a comprehensive methodological review emphasizing rarely discussed results and the interdisciplinary relevance of integral invariants.

Findings

Links integral invariants to diverse physical theories

Highlights rarely expounded results in the theory

Connects historical development with modern applications

Abstract

In this methodological review, we discuss the fundamental concepts of the theory of integral invariants. This theory originated with Poincare and Cartan \cite{Koz, Kart} and was further developed by Kozlov \cite{int_K}. We demonstrate how the core ideas of this theory link diverse fields of mathematical physics, such as Hamiltonian dynamics, optics, and hydrodynamics. Particular attention is paid to results that are rarely expounded in standard textbooks.