Symmetries and invariants of the time-dependent oscillator equation and the envelope equation

Hong Qin; Ronald C. Davidson

doi:10.1109/PAC.2005.1591095

Symmetries and invariants of the time-dependent oscillator equation and the envelope equation

Hong Qin, Ronald C. Davidson

PPPL Computational Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Abstract

The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant is fundamentally a result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency. A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.

Original language	English (US)
Title of host publication	Proceedings of the Particle Accelerator Conference, PAC 2005
Pages	2315-2317
Number of pages	3
DOIs	https://doi.org/10.1109/PAC.2005.1591095
State	Published - 2005
Event	Particle Accelerator Conference, PAC 2005 - Knoxville, TN, United States Duration: May 16 2005 → May 20 2005

Publication series

Name	Proceedings of the IEEE Particle Accelerator Conference
Volume	2005

Conference

Conference	Particle Accelerator Conference, PAC 2005
Country/Territory	United States
City	Knoxville, TN
Period	5/16/05 → 5/20/05

All Science Journal Classification (ASJC) codes

General Engineering

Access to Document

10.1109/PAC.2005.1591095

Cite this

@inproceedings{a37d50f63da0432787b666b6100b8cbd,

title = "Symmetries and invariants of the time-dependent oscillator equation and the envelope equation",

abstract = "The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant is fundamentally a result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency. A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.",

author = "Hong Qin and Davidson, \{Ronald C.\}",

year = "2005",

doi = "10.1109/PAC.2005.1591095",

language = "English (US)",

isbn = "0780388593",

series = "Proceedings of the IEEE Particle Accelerator Conference",

pages = "2315--2317",

booktitle = "Proceedings of the Particle Accelerator Conference, PAC 2005",

note = "Particle Accelerator Conference, PAC 2005 ; Conference date: 16-05-2005 Through 20-05-2005",

}

Qin, H & Davidson, RC 2005, Symmetries and invariants of the time-dependent oscillator equation and the envelope equation. in Proceedings of the Particle Accelerator Conference, PAC 2005., 1591095, Proceedings of the IEEE Particle Accelerator Conference, vol. 2005, pp. 2315-2317, Particle Accelerator Conference, PAC 2005, Knoxville, TN, United States, 5/16/05. https://doi.org/10.1109/PAC.2005.1591095

Symmetries and invariants of the time-dependent oscillator equation and the envelope equation. / Qin, Hong; Davidson, Ronald C.
Proceedings of the Particle Accelerator Conference, PAC 2005. 2005. p. 2315-2317 1591095 (Proceedings of the IEEE Particle Accelerator Conference; Vol. 2005).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

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T1 - Symmetries and invariants of the time-dependent oscillator equation and the envelope equation

AU - Qin, Hong

AU - Davidson, Ronald C.

PY - 2005

Y1 - 2005

N2 - The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant is fundamentally a result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency. A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.

AB - The single-particle dynamics in a time-dependent focusing field is examined. The existence of the Courant-Snyder invariant is fundamentally a result of the corresponding symmetry admitted by the oscillator equation with time-dependent frequency. A careful analysis of the admitted symmetries reveals a deeper connection between the nonlinear envelope equation and the oscillator equation. A general theorem regarding the symmetries and invariants of the envelope equation, which includes the existence of the Courant-Snyder invariant as a special case, is demonstrated. The symmetries of the envelope equation enable a fast algorithm for finding matched solutions without using the conventional iterative shooting method.

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M3 - Conference contribution

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SN - 0780388593

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T3 - Proceedings of the IEEE Particle Accelerator Conference

SP - 2315

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BT - Proceedings of the Particle Accelerator Conference, PAC 2005

T2 - Particle Accelerator Conference, PAC 2005

Y2 - 16 May 2005 through 20 May 2005

ER -

Symmetries and invariants of the time-dependent oscillator equation and the envelope equation

Abstract

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Conference

All Science Journal Classification (ASJC) codes

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