Symmetries and Integrals in Mechanics

Abstract

This chapter discusses the concept of symmetries and integrals in mechanics. The literature of Hamiltonian mechanics has several special theorems dealing with Hamiltonian systems that admit additional integrals in involution. Two examples are: (1) If a Hamiltonian system admits p additional independent integral in involution, then the algebraic multiplicity of + 1 as a characteristic multiplier of a periodic solution is greater than or equal to 2(p + 1); and (2) the integration of a Hamiltonian system of n degrees of freedom that admits p independent integrals in involution can be reduced to the integration of a Hamiltonian system of n − p degrees of freedom with p parameters and additional quadratures. After restating (2) in modern terminology, these theorems are generalized by dropping the assumption that the integrals are in involution.