Advanced Classical Mechanics/Poisson Brackets

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Poisson Brackets[edit | edit source]

The Poisson bracket of any two functions, and , is:

In two dimensions, the multivariable chain rule, is . Using implied summation notation (for the index j), we apply this to Hamilton's equations:

As an aside, we note a connection to Quantum Mechanics: Ehrenfest theorem involves the operators and expectation values of quantum mechanics. It states: [1]

where is any operator of quantum mechanics, is its expectation value, and

is the commutator of and .

References[edit | edit source]

  1. Smith, Henrik (1991). Introduction to Quantum Mechanics. World Scientific Pub Co Inc. pp. 108–109. ISBN 978-9810204754.