Chapter

Hamilton-Jacobi Theory

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print June 2018 | ISBN: 9780198822370
Published online August 2018 | e-ISBN: 9780191861253 | DOI: https://dx.doi.org/10.1093/oso/9780198822370.003.0019
Hamilton-Jacobi Theory

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This chapter focuses on Liouville’s theorem and classical statistical mechanics, deriving the classical propagator. The terms ‘phase space volume element’ and ‘Liouville operator’ are defined and an n-particle phase space probability density function is constructed to derive the Liouville equation. This is deconstructed into the BBGKY hierarchy, and radial distribution functions are used to develop n-body correlation functions. Koopman–von Neumann theory is investigated as a classical wavefunction approach. The chapter develops an operatorial mechanics based on classical Hilbert space, and discusses the de Broglie–Bohm formulation of quantum mechanics. Partition functions, ensemble averages and the virial theorem of Clausius are defined and Poincaré’s recurrence theorem, the Gibbs H-theorem and the Gibbs paradox are discussed. The chapter also discusses commuting observables, phase–amplitude decoupling, microcanonical ensembles, canonical ensembles, grand canonical ensembles, the Boltzmann factor, Mayer–Montroll cluster expansion and the equipartition theorem and investigates symplectic integrators, focusing on molecular dynamics.

Keywords: Liouville’s theorem; Liouville operator; classical propagator; BBGKY hierarchy; radial distribution function; n-body correlation function; Koopman–von Neumann theory; classical wavefunction; classical statistical mechanics; ensemble average

Chapter.  9343 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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