dorsal/arxiv
View SchemaThe Adiabatic Invariance of the Action Variable in Classical Dynamics
| Authors | Clive G. Wells, Stephen T. C. Siklos |
|---|---|
| Categories | |
| ArXiv ID | physics/0610084 |
| URL | https://arxiv.org/abs/physics/0610084 |
| DOI | 10.1088/0143-0807/28/1/011 |
| Journal | Eur. J. Phys, 28, 105-112, 2007. |
Abstract
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.
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"abstract": "We consider one-dimensional classical time-dependent Hamiltonian systems with\nquasi-periodic orbits. It is well-known that such systems possess an adiabatic\ninvariant which coincides with the action variable of the Hamiltonian\nformalism. We present a new proof of the adiabatic invariance of this quantity\nand illustrate our arguments by means of explicit calculations for the harmonic\noscillator.\n The new proof makes essential use of the Hamiltonian formalism. The key step\nis the introduction of a slowly-varying quantity closely related to the action\nvariable. This new quantity arises naturally within the Hamiltonian framework\nas follows: a canonical transformation is first performed to convert the system\nto action-angle coordinates; then the new quantity is constructed as an action\nintegral (effectively a new action variable) using the new coordinates. The\nintegration required for this construction provides, in a natural way, the\naveraging procedure introduced in other proofs, though here it is an average in\nphase space rather than over time.",
"arxiv_id": "physics/0610084",
"authors": [
"Clive G. Wells",
"Stephen T. C. Siklos"
],
"categories": [
"physics.class-ph"
],
"doi": "10.1088/0143-0807/28/1/011",
"journal_ref": "Eur. J. Phys, 28, 105-112, 2007.",
"title": "The Adiabatic Invariance of the Action Variable in Classical Dynamics",
"url": "https://arxiv.org/abs/physics/0610084"
},
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