Integral Invariants and Hamiltonian Systems
Oleg Zubelevich
TL;DR
This article surveys integral invariants rooted in Poincaré and Cartan and reveals their unifying role across Hamiltonian dynamics, optics, and hydrodynamics. By formulating invariants as differential forms and exploiting the Lie derivative, it connects conservation laws, symplectic structure, and geometric flows, introducing the Poincaré–Cartan invariant $\alpha=p_i\,dx^i-H\,dt$ and its relatives. It develops nonautonomous extensions, Darboux normal forms, and canonical transformations with generating functions, illustrating how energy-level cuts, Hamilton–Jacobi theory, and the method of characteristics yield practical reductions and closed-form integrations in diverse settings. The text emphasizes concrete, often underrepresented results, such as time-dependent invariants, Hamiltonian straightening, and Poincaré cuts, which together provide a cohesive geometric framework for analyzing Hamiltonian systems and their reductions.
Abstract
In this review and methodological article we discuss the main ideas of the integral invariants theory. This theory was originated by Poincare and Cartan. We show how ideas of this theory connect such different fields of mathematical physics as Hamiltonian dynamics, optics and hydrodynamics. We focus our attention on the results that are rarely expounded in the textbooks.