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Dynamical symmetries of the anisotropic oscillator

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Authors:
Akash Sinha at Indian Institute of Technology Bhubaneswar
Akash Sinha
Aritra Ghosh at Indian Institute of Technology Bhubaneswar
Aritra Ghosh
Bijan Bagchi
Bijan Bagchi
Bijan Bagchi
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Abstract

It is well known that the Hamiltonian an n-dimensional isotropic oscillator admits of an SU(n) symmetry, making the system maximally superintegrable. However, the dynamical symmetries of the anisotropic oscillator are much more subtle. We introduce a novel set of canonical transformations that map an n-dimensional anisotropic oscillator to the corresponding isotropic problem. Consequently, the anisotropic oscillator is found to possess the same number of conserved quantities as the isotropic oscillator, making it maximally superintegrable too. The first integrals are explicitly calculated in the case of a two-dimensional anisotropic oscillator and remarkably, they admit closed-form expressions.
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Phys. Scr. 98 (2023)095253 https://doi.org/10.1088/1402-4896/acf00d
PAPER
Dynamical symmetries of the anisotropic oscillator
Akash Sinha
1
, Aritra Ghosh
1,∗
and Bijan Bagchi
2
1
School of Basic Sciences, Indian Institute of Technology Bhubaneswar, Jatni, Khurda, Odisha 752050, India
2
Shiv Nadar University, Physics Dept., Gautam Buddha Nagar, Uttar Pradesh 203207, India
∗
Author to whom any correspondence should be addressed.
E-mail: s23ph09005@iitbbs.ac.in,ag34@iitbbs.ac.in and bbagchi123@gmail.com
Keywords: anisotropic oscillator, symmetries, unitary group
Abstract
It is well known that the Hamiltonian of an n-dimensional isotropic oscillator admits an SU(n)
symmetry, making the system maximally superintegrable. However, the dynamical symmetries of the
anisotropic oscillator are much more subtle. We introduce a novel set of canonical transformations
that map an n-dimensional anisotropic oscillator to the corresponding isotropic problem.
Consequently, the anisotropic oscillator is found to possess the same number of conserved quantities
as the isotropic oscillator, making it maximally superintegrable too. The first integrals are explicitly
calculated in the case of a two-dimensional anisotropic oscillator and remarkably, they admit closed-
form expressions.
1. Introduction
The harmonic oscillator plays a pivotal role in our understanding of various physical phenomena, starting from
classical dynamical systems to those in quantum field theories. It admits a Hamiltonian description as given by
the equations of motion:
{} ()=
=-
ÎqH
ppH
qjn,,1,2,,, 1
j
j
j
j
where His quadratic in npairs of (q
j
,p
j
), in the phase space defined for ndegrees of freedom. For the simplest
case of the isotropic oscillator, the Hamiltonian reads
() ()
åw=+
=
Hpq
1
2,2
j
n
jj
1
2
0
22
which is a linear sum, i.e.
=å
=
HH
j
nj
1
, with H
j
being the component Hamiltonian of the jth oscillator having
coordinate q
j
, momentum p
j
, and frequency (independent of j)ω
0
. Thus, the system executes simple harmonic
motion on each q
j
−p
j
plane in the phase space with the same frequency ω
0
. As is well known, the system is
superintegrable [1], meaning that it possesses more functionally independent constants of motion than what is
required for Liouville-Arnold integrability [2]
3
. The symmetry group of the Hamiltonian is SU(n)[4,5], and the
conserved quantities follow the su(n)Lie algebra, with respect to the Poisson bracket. It means that all the
conserved quantities are not in involution, i.e. they do not admit mutually commuting Poisson brackets. This is
certainly expected of any superintegrable system.
4
As a matter of fact, the isotropic oscillator is maximally
superintegrable, meaning that it possesses the maximum number of independent integrals of motion that a
system with a 2n-dimensional phase space can admit.
For concreteness, we focus on the two-dimensional (n=2)case which corresponds to a four-dimensional
phase space. Then, the Hamiltonian is
RECEIVED
12 May 2023
REVISED
3 August 2023
ACCEPTED FOR PUBLICATION
14 August 2023
PUBLISHED
29 August 2023
3
For a general classification of classical superintegrable systems with the help of ladder operators, see [3].
4
It should be noted that any integrable Hamiltonian system is superintegrable in the neighborhood of any regular point in the classical phase
space (see for example [6]).
© 2023 IOP Publishing Ltd
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