Calogero Moser system - davidar/scholarpedia GitHub Wiki
Calogero-Moser dynamical system is a one-dimensional many-body problem that can be explicitly solved.
The dynamical system that is generally called Calogero-Moser (hereafter CM) is the model characterized by the Hamiltonian
- <math>\label{Ham}
Here and hereafter <math>N</math> is an arbitrary positive integer, the two (real) <math>N</math>-vectors <math>\underline{p}\equiv \left( p_{1},...,p_{N}\right) </math> [respectively]<math>\underline{q}\equiv \left( q_{1},...,q_{N}\right)] </math> feature as their components the <math>N</math> canonical momenta <math>p_{n}</math> [respectively], <math>g^{2}</math> is a positive "coupling constant" characterizing the strength of the interparticle two-body interaction and <math>\omega^{2}</math> is a nonnegative constant characterizing the strength of the interaction with an external "harmonic oscillator" potential. This Hamiltonian describes the (nonrelativistic) one-dimensional <math>N</math>-body problem of <math>N</math> equal particles (whose mass has been set to unity) interacting pairwise via a repulsive force singular at zero separation and with a common, confining, external "harmonic oscillator" potential (absent if <math>\omega =0</math>). An analogous model is characterized by the Hamiltonian
- <math>\label{Ham2}
It has the merit – in contrast to \eqref{Ham} – of being translation-invariant, and yet to yield, in its center-of-mass system, an almost identical dynamics to that yielded by the Hamiltonian \eqref{Ham}: the difference among the two models is that the center-of-mass oscillates harmonically in the first case, \eqref{Ham}, while it moves freely in the second case, \eqref{Ham2}.
In the classical context the Newtonian equations of motion yielded by \eqref{Ham} read
- <math>\label{Newt}
where of course <math>q_{n}\equiv q_{n}\left( t\right) \ ,</math> the (real) independent variable <math>t</math> is the time and superimposed dots denote time differentiations.
In the quantal context the stationary Schrödinger equation corresponding to the Hamiltonian \eqref{Ham} reads
- <math>\label{Schr}
where <math>\Delta </math> is the Laplace operator in the <math>N</math>-dimensional space spanned by the <math>N</math> coordinates <math>x_{n}\ ,</math> <math>\Delta =\sum_{n=1}^{N}\partial^{2}/\partial x_{n}^{2}\ ,</math> and <math>\Psi \equiv \Psi \left(x_{1},...,x_{N}\right) </math> is the eigenfunction corresponding to the energy eigenvalue <math>E\ ;</math> note that we set to unity the Planck constant, <math>\hbar =1\ .</math>
The interest of this model lies in its exact solvability, both in the classical and quantal contexts.
In the classical context, the Hamiltonian model \eqref{Ham} is completely integrable – <math>N</math> integrals of motion in involution can be explicitly exhibited – indeed superintegrable – <math>2N-1</math> functionally independent integrals of motion can be explicitly exhibited – and algebraically solvable: the solution of the initial-value problem of the Newtonian equations of motion \eqref{Newt} can be performed by algebraic operations, specifically by computing the <math>N</math> eigenvalues of an <math>N\times N</math> matrix explicitly known in terms of the initial data <math>q_{n}\left( 0\right) \ ,</math> <math>\dot{q}_{n}\left( 0\right) </math> and of the time <math>t\ .</math> In the confined case (<math>\omega >0</math>), the solution is isochronous: completely periodic,
- <math>
- <math>\label{T}
In the not confined case (<math>\omega =0</math>) the time evolution of the <math>N</math> particles features the following neat asymptotic relation among their positions and velocities <math>p_{n}=\dot{q}_{n}\ ,</math> in the remote past and future:
- <math>\label{Asy}
- <math>
- <math>\label{Asyc}
with the ordering of these velocities corresponding of course to the ordering of the particles on the line, say <math>p_{1}^{\left( -\right)}>p_{2}^{\left( -\right) }>...>p_{N}^{\left( -\right) }</math> and <math></math>p_{1}^{\left( +\right) }<p_{2}^{\left(
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=" [BC1990]"></p_{2}^{\left(>II, Phys. Rev. A4 (1971), 2019-2021 (1971) & A5(1972), 1372-1376
[T1967] M. Toda: Vibration of a chain with a nonlinear interaction, J. Phys. Soc. Japan 22 (1967), 431-436
Internal references
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- James Meiss (2007) Hamiltonian systems. Scholarpedia, 2(8):1943.
- Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
- Martin Gutzwiller (2007) Quantum chaos. Scholarpedia, 2(12):3146.
- David H. Terman and Eugene M. Izhikevich (2008) State space. Scholarpedia, 3(3):1924.
- Perelomov, A. M. 1990. Integrable systems of classical mechanics and Lie algebras. Birkhauser, Basel.
- Hoppe, J. 1992. Lectures on Integrable Systems. Springer, Berlin.
- Calogero, F. 2001. Classical many-body problems amenable to exact treatments, Lecture Notes in Physics Monograph m66. Springer, Berlin.
- Babelon, O., Bernard D. and Talon M. 2003. Introduction to classical integrable systems. Cambridge University Press, Cambridge.
- Etingof, P. 2007. Calogero-Moser Systems and Representation Theory. Zurich Lectures in Advanced Mathematics, European Mathematical Society Publishing House.
- Calogero, F. 2008. Isochronous systems. Oxford University Press, Oxford.