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.
KeywordsPoisson Bracket Symplectic Form Symplectic Manifold Poisson Structure Jacobi Identity
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