Advanced Classical Mechanics/Poisson Brackets

From Wikiversity
Jump to navigation Jump to search
Nuvola apps edu mathematics-p.svg Subject classification: this is a mathematics resource.

Poisson Brackets[edit]

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]

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