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Introduction

  • Maciej Błaszak
Chapter

Abstract

Classical and quantum Hamiltonian mechanics belong to the most important physical theories which are able to model with an incredible precision various physical processes which take place in the real world, from astronomical macro scale to atomic and molecular micro scale. Historically, classical Hamiltonian mechanics grew out from Newtonian (later on Lagrangian) mechanics, describing particle dynamics under influence of potential forces, in the form of second order ordinary differential equations (ODE’s) in base Euclidian (Riemannian) space. Simple, n second order ODE’s on a base space was replaced by 2n first order ODE’s on a phase space, parametrized by n position coordinates and n momentum coordinates. In such formulation, the flow, governed by conserved total energy (classical Hamiltonian) of the system, represented particle dynamics on the phase space. Such Hamiltonians consisted of the kinetic part, quadratic in momenta, and the potential part, position dependent. Since then, the Hamiltonian mechanics has developed into an independent general theory allowing to describe a much wider class of dynamical systems than only particle dynamics on some configuration space (base space). Actually, it is a theory of Hamiltonian flows on Poisson manifolds M, governed by arbitrary smooth real valued functions (Hamiltonians) on M. In consequence, considered dynamical systems are subject to Poisson geometry. Obviously, for particular Poisson manifolds and particular Hamiltonians, one can adopt the Riemannian geometry to Hamiltonian formalism, regarding a Poisson manifold as a cotangent bundle to some Riemannian space and momentum part of Hamiltonian as defined by a respective metric tensor. Nevertheless, on the general level of the Hamiltonian formalism, there is no related Riemannian geometry and hence there is no configuration space where the dynamics could be transferred.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Maciej Błaszak
    • 1
  1. 1.Division of Mathematical PhysicsFaculty of Physics UAMPoznańPoland

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