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The Lazy UniverseAn Introduction to the Principle of Least Action$
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Jennifer Coopersmith

Print publication date: 2017

Print ISBN-13: 9780198743040

Published to Oxford Scholarship Online: June 2017

DOI: 10.1093/oso/9780198743040.001.0001

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Hamiltonian Mechanics

Hamiltonian Mechanics

(p.142) (p.143) 7 Hamiltonian Mechanics
The Lazy Universe

Jennifer Coopersmith

Oxford University Press

Hamilton’s genius was to understand what were the true variables of mechanics (the “pq,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “pq” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

Keywords:   Hamiltonian, phase space, Hamilton-Jacobi, conjugate coordinates, canonical equations, infinitesimal canonical transformations, geometrical optics, Huygen’s Principle, Schrödinger, symmetry

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