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Fundamental concepts

  • Richard H. Cushman
  • Larry M. Bates
Chapter

Abstract

In this chapter we describe the basic mathematical structures needed to do Hamiltonian mechanics. We begin with a section on symplectic linear algebra. The motion of a Hamiltonian system takes place on a symplectic manifold, that is, a manifold with a closed nondegenerate 2-form, called a symplectic form. The symplectic form allows one to turn the differential of a function, called a Hamiltonian, into a vector field whose integral curves satisfy Hamilton’s equations. An algebraic way of treating Hamiltonian mechanics is via Poisson brackets. When the vector space of smooth functions on a symplectic manifold, which is a Lie algebra under Poisson bracket, is made into an algebra using pointwise multiplication of smooth functions, we obtain a Poisson algebra. The symplectic formulation of mechanics can be recovered from this Poisson algebra.

Keywords

Poisson Bracket Symplectic Form Symplectic Manifold Poisson Structure Jacobi Identity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada
  2. 2.University of CalgaryCalgaryCanada

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