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Integral Invariants

  • Alexei Deriglazov
Chapter

Abstract

This chapter is devoted to the discussion of the theory of integral invariants, which reveals an interesting structure of the general solution to Hamiltonian equations. We discuss the basic Poincaré-Cartan and Poincaré integral invariants that represent line integrals of a special vector field defined on extended phase space. The integrals retain the same value for any closed contour taken on a given two-dimensional surface formed by solutions to the Hamiltonian equations. As will be discussed in Sect. 5.1.3, this property could be taken as a basic principle of mechanics, instead of the principle of least action. Besides their applications in mechanics, integral invariants are widely used in the theory of differential equations, see [1, 4].

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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  • Alexei Deriglazov
    • 1
    • 2
  1. 1.Universidade Federal de Juiz de Fora Dept. of MathematicsJuiz de ForaBrazil
  2. 2.Laboratory of Mathematical PhysicsTomsk Polytechnic UniversityTomskRussian Federation

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