View source for Ruijsenaars-Schneider model

===Classical level===

The defining Hamiltonian is given by
:<math>
H= mc^2 \sum_{j=1}^{N}\cosh(\frac{p_j}{mc})\prod_{1\le k\le N,k\ne
j}f(x_j-x_k).
</math>
The special case <math> f(x)=1</math> corresponds to <math>N</math> particles with positions <math>x_1,\ldots,x_N</math> and rapidities <math>p_1/mc,\ldots,p_N/mc\ ,</math> moving freely along the line. The generalized pair potential
:<math>
f^2(x)=a+b\wp(x)
</math>
(with <math>a,b</math> positive constants and <math>\wp(x)</math> the '''Weierstrass <math>\wp</math>-function''') is the most general interaction compatible with relativistic invariance, important special cases being
:<math>
f^2(x) = \left\{
\begin{array}{ll}
1+g^2/m^2c^2x^2  &
({\rm rational})  \\
1+\sin^2(\nu g/mc)/
\sinh^2(\nu x)  &   ({\rm hyperbolic})  \\
1+\sinh^2(\nu g/mc)/
\sin^2(\nu x)  &   ({\rm trigonometric}).
\end{array}
\right.
 </math>
More specifically, introducing 
:<math>
P= mc \sum_{j=1}^{N}\sinh(\frac{p_j}{mc})\prod_{1\le k\le N,k\ne
j}f(x_j-x_k),
</math>
and
:<math>
B=-m\sum_{j=1}^Nx_j,
</math>
one obtains Poisson brackets
:<math>
\{ H,P\}=0,\  \ \{ H,B\} =P, \ \  \{ P,B\} =H/c^2,
</math>
which represent the Lie algebra of the inhomogeneous Lorentz group. Thus, <math>H,P</math> and <math>B</math> represent the time translation, space translation and Lorentz boost generator, respectively. The second and third bracket hold true for an arbitrary <math>f\ ,</math> but the first one (encoding space-time translation invariance) yields a severe constraint on <math>f^2</math> (for <math>N>2</math>): Its vanishing amounts to functional equations for <math>f^2</math> that hold true for the Weierstrass <math>\wp</math>-function.

For the above three special cases one easily checks the nonrelativistic limits
:<math>
\lim_{c\rightarrow \infty }(H-Nmc^2)=H_{\rm nr},
 \ \ \ \lim_{c\rightarrow \infty}P=P_{\rm nr},
</math>
where
:<math>
H_{\rm nr}=\frac{1}{2m}\sum_{j=1}^N p_j^2 + g^2
\sum_{1\le j<k\le N} V(x_j-x_k),\ \ \ P_{\rm nr}=\sum_{j=1}^Np_j,
</math>
with
:<math>
V(x) = \left\{
\begin{array}{ll}
1/mx^2  &
({\rm rational})  \\
\nu^2/
m\sinh^2(\nu x)  &   ({\rm hyperbolic})  \\
\nu^2/
m\sin^2(\nu x)  &   ({\rm trigonometric}).
\end{array}
\right.
 </math>
With a suitable choice of <math>a,b</math> for the elliptic case, one again gets these limits, now with
:<math>
V(x)=\wp(x)/m.
</math>
Furthermore, the limits of the Poisson brackets, namely,
:<math>
\{ H_{\rm nr},P_{\rm nr}\}=0,\  \ \{ H_{\rm nr},B\} =P_{\rm nr}, \ \  \{ P_{\rm nr},B\} =Nm,
</math>
represent the Lie algebra of the nonrelativistic space-time symmetry group (the Galilei group).

The Hamiltonian <math>H_{\rm nr}</math> is the defining Hamiltonian of the nonrelativistic Calogero-Moser system. One obtains a representation of the Galilei group Lie algebra for any (even) <math>V(x)\ ,</math> but the <math>\wp</math>-function choice is essential for <math>H_{\rm nr}</math> to give rise to an integrable system. For <math>H</math> and <math>P\ ,</math> by contrast, this choice is essential to obtain translation invariance, but as a bonus it preserves integrability. Specifically, it is clear that the functions
:<math>
S_{\pm N}=\exp(\pm \beta\sum_{j=1}^Np_j),\ \ \ \beta=1/mc
</math>
Poisson commute with <math>H</math> and <math>P\ ,</math> which already yields integrability for <math>N=3\ .</math> More generally, 
it can be shown that the Hamiltonians
:<math>
S_{\pm l}=\sum_{\stackrel{\scriptstyle
I\subset \{ 1,\ldots ,N\} }{|I|=l}}\exp (\pm \beta
\sum_{j\in I}p_j )\prod_{\stackrel{\scriptstyle
j\in I}{k\not\in I}}f(x_j-x_k),\ \ \ l=1,\ldots,N
</math>
mutually commute. Observe that one has
:<math>
S_{-l}=S_{-N}S_{N-l},\ \ \ l=1,\ldots,N-1,
</math>
and
:<math>
H=(S_1+S_{-1})/2m\beta^2,\ \ \ P=(S_1-S_{-1})/2\beta.
</math>