Abstract

On Applications of Group Analysis of Differential Equations in Analytical Dynamics. A Fragment of University Special Course

I.S.Yemelyanova (Nizhny Novgorod State University, Nizhny Novgorod, Russia)

e-mail: root@yemel.sci-nnov.ru

A course "Group analysis of differential equations and its applications'' [1,2] for Nizhny Novgorod State University students and postgraduates contains group theoretical methods in analytical mechanics. The following topics are discussed: Lagrange equations and classical laws of conservation; noncovariance of ignorable coordinate definition for holonomic and nonholonomic systems; covariant generation of Lagrange equations using Lie derivative; Emmy Noether theorem and its canonical form; the main methods of integrability for canonical equations and their group theoretical nature (Poisson method and general Poisson brackets, canonical transformations, Hamilton -- Jacoby equation, method of small perturbations and Lie group estimations for the small perturbations method, integral invariants, adiabatic invariants); Hamilton symmetry; the main types of symmetries for Hamiltonian systems (Dynamical, Lie, Lie prolonged, Cartan, Noether, Hamiltonian symmetries) and their comparison). Bibliography: 1. Yemelyanova I.S., Symmetries and constants of the motion for analytical dynamics. Nizhny Novgorod: Univ. Press, 1992 (in Russian). 2. Yemelyanova I.S. Group analysis of differential equations. Tasks and examples. Nizhny Novgorod: Univ. Press, 2001 (in Russian).

Section : 12