Journal of Nonlinear Mathematical Physics

Volume 17, Issue 1, March 2010, Pages 103 - 110

Isochronous Oscillators

Authors
F. Calogero
Dipartimento di Fisica, Università di Roma “La Sapienza”, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Roma, francesco.calogero@roma1.infn.it, francesco.calogero@uniroma1.it
F. Leyvraz
Centro Internacional de Ciencias, Cuernavaca, Mexico
Departamento de Física, Universidad de los Andes, Bogotá, Colombia, fa.leyvraz44@uniandes.edu.co
Received 12 May 2009, Accepted 11 August 2009, Available Online 7 January 2021.
DOI
https://doi.org/10.1142/S1402925110000611How to use a DOI?
Keywords
Nonlinear oscillators, isochronous Hamiltonians, quantization, equispaced spectrum, infinite degeneracy
Abstract

We exhibit the solution of the initial-value problem for the system of 2N + 2 oscillators characterized by the Hamiltonian

H(p^0,pˇ0,p^_,pˇ_,q^0,qˇ0,q^_,qˇ_)=12[p^02pˇ02+Ω2(q^02qˇ02)]+q^0Ωqˇ02bn=1N[p^n2pˇn2+ωn2(q^n2qˇn2)]+p^0Ωq^0bn=1N[p^npˇn+ωn2q^nqˇn]
where N is an arbitrary positive integer, Ω, b and ωn2 are N + 2 arbitrary real constants, q^m,qˇm with m = 0,1,…,N are the 2N + 2 canonical coordinates and p^m,pˇm the corresponding 2N + 2 canonical momenta. In the classical context the solution is completely periodic with period T = 2π/|Ω| (for arbitrary initial data). In the quantal context the (infinitely degenerate) spectrum is equispaced, with spacing ħ|Ω|; all the corresponding eigenfunctions are also exhibited. This finding obtains as special case of a more general (new) class of isochronous Hamiltonians.

Copyright
© 2010 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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Journal
Journal of Nonlinear Mathematical Physics
Volume-Issue
17 - 1
Pages
103 - 110
Publication Date
2021/01
ISSN (Online)
1776-0852
ISSN (Print)
1402-9251
DOI
https://doi.org/10.1142/S1402925110000611How to use a DOI?
Copyright
© 2010 The Authors. Published by Atlantis Press and Taylor & Francis
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - F. Calogero
AU  - F. Leyvraz
PY  - 2021
DA  - 2021/01
TI  - Isochronous Oscillators
JO  - Journal of Nonlinear Mathematical Physics
SP  - 103
EP  - 110
VL  - 17
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925110000611
DO  - https://doi.org/10.1142/S1402925110000611
ID  - Calogero2021
ER  -