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ERIC Number: EJ827976
Record Type: Journal
Publication Date: 2007-Jan
Pages: 8
Abstractor: As Provided
Reference Count: 2
ISSN: ISSN-0143-0807
The Adiabatic Invariance of the Action Variable in Classical Dynamics
Wells, Clive G.; Siklos, Stephen T. C.
European Journal of Physics, v28 n1 p105-112 Jan 2007
We consider one-dimensional classical time-dependent Hamiltonian systems with quasi-periodic orbits. It is well known that such systems possess an adiabatic invariant which coincides with the action variable of the Hamiltonian formalism. We present a new proof of the adiabatic invariance of this quantity and illustrate our arguments by means of explicit calculations for the harmonic oscillator. The new proof makes essential use of the Hamiltonian formalism. The key step is the introduction of a slowly varying quantity closely related to the action variable. This new quantity arises naturally within the Hamiltonian framework as follows: a canonical transformation is first performed to convert the system to action-angle coordinates; then the new quantity is constructed as an action integral (effectively a new action variable) using the new coordinates. The integration required for this construction provides, in a natural way, the averaging procedure introduced in other proofs, though here it is an average in phase space rather than over time. (Contains 5 footnotes.)
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Publication Type: Journal Articles; Reports - Research
Education Level: Higher Education
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A