Ruijsenaars–Schneider models (short RS models) are

relativistic Relativity may refer to: Physics * Galilean relativity, Galileo's conception of relativity * Numerical relativity, a subfield of computational physics that aims to establish numerical solutions to Einstein's field equations in general relativity ...

generalizations of Calogero–Moser–Sutherland models (short CMS models), which are closely connected with relativistic field theories like the

Sine-Gordon model The sine-Gordon equation is a second-order nonlinear partial differential equation for a function \varphi dependent on two variables typically denoted x and t, involving the wave operator and the sine of \varphi. It was originally introduced by ...

. RS models are named after Simon N. M. Ruijsenaars and Herbert Schneider. Both of them introduced the classical models in 1986 and Ruijsenaars introduced the

quantum In physics, a quantum (: quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a property can be "quantized" is referred to as "the hypothesis of quantization". This me ...

models in 1987.

Description

For

N

particles on the real line

\mathbb

, the Hamiltonian of the RS model is given by: :

H
=mc^2\sum_^N\cosh\left(\frac\right)\prod_f(x_i-x_j).

Different

potentials Potential generally refers to a currently unrealized ability, in a wide variety of fields from physics to the social sciences. Mathematics and physics * Scalar potential, a scalar field whose gradient is a given vector field * Vector potential ...

lead to different RS models, which the four types most often considered being: * Type I/rational: *:

f(x)^2
=1+\left(\frac\right)^2.

* Type II/hyperbolic: *:

f(x)^2
=1+\frac.

* Type III/trigonometric: *:

f(x)^2
=1+\frac.

* Type IV/elliptic:Ruijsenaars 1987, Equation (1.4) *:

f(x)^2
=a+b\weierp(x;\omega_1,\omega_2).

Literature

* * * * {{cite arXiv , eprint=2312.12932 , class= math-ph, first=Martin , last=Hallnäs , title=Calogero-Moser-Sutherland systems , date=2023-12-20

References

Mathematical physics Dynamical systems