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{\noindent\Large\bf Deformations of Calogero-Moser Systems}\footnote{Talk
given at the 9th Workshop on {\sl Nonlinear Evolution Equations and Dynamical
Systems \\ (NEEDS '93)}, held at Gallipoli, Italy, September 3-12, 1993.}
\vspace{3ex}
{\noindent\large J.F. van Diejen}\footnote{E-mail
address: jand@fwi.uva.nl}
{\noindent\small
Department of Mathematics and Computer Science, University of Amsterdam,\\
Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands.}
\vspace{1ex}
%\begin{quotation}
{\small\noindent {\bf Abstract.}
Recent results are surveyed pertaining to the complete integrability of
some novel $n$-particle models in dimension one.
These models generalize the Calogero-Moser systems
related to classical root systems.}
\vspace{3ex}
\noindent {\bf 1. Introduction} \hfill
\noindent The Hamiltonian of the celebrated Calogero-Moser (CM) system
\cite{op}
is given by
\begin{equation}\label{cm}
H_{cm}= 1/2 \sum_{1\leq j\leq n} \theta_j^2 +
g^2 \sum_{1\leq j < k\leq n} \wp (x_j-x_k) ,
\end{equation}
where $\wp (z)$ denotes the Weierstra\ss{} $\wp$-function \cite{ww}
or a degeneration thereof ($1/z^2$, $1/{\rm sh}^2(z)$ or $1/\sin^2 (z)$).
The integrability of $H_{cm}$ (\ref{cm}) was proved with the aid of a
Lax matrix \cite{op}.
Some years ago a relativistic generalization of $H_{cm}$ was
introduced \cite{rs,r1,r2}. The Hamiltonian of the relativistic system
(RCM) reads
\begin{equation}\label{rcm}
H_{rcm}=\sum_{1\leq j\leq n} {\rm ch} (\beta\theta_j)
\prod_{k\neq j} \left[ 1+ \beta^2g^2 \wp (x_j-x_k) \right]^{1/2} .
\end{equation}
One can look upon the RCM system as a one-parameter deformation of the
CM model, with $\beta \sim 1/c$ (the inverse of the speed of light)
acting as deformation parameter.
For $\beta \rightarrow 0$, which corresponds to the nonrelativistic
limit, $\beta^{-2}(H_{rcm}(\beta )-n)$ converges to $H_{cm}$.
The relativistic system is also integrable;
explicit formulas have been found for a complete set of
integrals in involution:
\begin{equation}
H_{l,\, rcm} =
\sum_{\stackrel{J\subset \{ 1,\ldots ,n\} }{|J|=l}}
e^{-\beta\sum_{j\in J}\theta_j}\,
\prod_{\stackrel{j\in J}{k\in J^c}}
\left[ 1+\beta^2g^2 \wp (x_j-x_k) \right]^{1/2},\;\;\;\;\;\;\;\;\;\;
l=1,\ldots ,n.
\end{equation}
{}From a Lie-theoretic perspective the above $n$-particle models are
connected with the root system $A_{n-1}$. Here, we will take a look at
similar deformations of the CM systems related to
classical root systems other than $A_{n-1}$ (i.e. $B_n$, $C_n$, $D_n$ and
$BC_n$).
A more detailed discussion of the material covered below
(including proofs) can be found in the papers \cite{die1,die2}.
\vspace{2ex}
\noindent {\bf 2. Trigonometric Potentials} \hfill
\noindent In the case of trigonometric potentials our system is characterized
by
the Hamiltonian
\begin{equation}
H = \sum_{1\leq j\leq n}\left( {\rm ch} (\beta \theta_j)\,
V_j^{1/2}V_{-j}^{1/2}\, -\,
(V_j+V_{-j})/2 \right) \label{Htr}
\end{equation}
with
\begin{eqnarray}\label{V1}
V_{\varepsilon j} &=& w(\varepsilon x_j)
\prod_{k\neq j} v(\varepsilon x_j+x_k) v(\varepsilon x_j-x_k),\;\;\;\;\;\;\;\;
\varepsilon =\pm 1, \\
v(z) &=& \frac{\sin\alpha (\mu +z)}{\sin (\alpha z)}, \\
w(z) &=& \frac{\sin\alpha (\mu_0 +z)}{\sin (\alpha z)}
\frac{\cos\alpha (\mu_1 +z)}{\cos (\alpha z)}
\frac{\sin\alpha (\mu_0^\prime +z)}{\sin (\alpha z)}
\frac{\cos\alpha (\mu_1^\prime +z)}{\cos (\alpha z)}
\label{trig} .
\end{eqnarray}
One can look upon the functions $v$ and $w$ as potentials:
$v$ governs the interaction between the particles and $w$ models an external
field.
The parameters $\mu$, $\mu_r$ and $\mu_r^\prime$ ($r=0,1$) act as
coupling constants; after setting them equal to zero the particles become
free ($v,w=1$).
Just as for the RCM system, explicit formulas have been found
that constitute a complete set of integrals in involution for the
Hamiltonian $H$~(\ref{Htr})-(\ref{trig})
\begin{equation}
H_l =
\sum_{\stackrel{J\subset \{ 1,\ldots ,n\} ,\, |J|\leq l}
{\varepsilon_j=\pm 1,\, j\in J} }
{\rm ch} (\beta \theta_{\varepsilon J})\,
V_{\varepsilon J;\, J^c}^{1/2}\, V_{-\varepsilon J;\, J^c}^{1/2}\, U_{J^c,\,
l-|J|},
\;\;\;\;\;\;\;\; l=1,\ldots ,n,
\end{equation}
with
\begin{eqnarray}
\theta_{\varepsilon J}&=& \sum_{j\in J}\; \varepsilon_j\, \theta_j, \\
V_{\varepsilon J;\, K} &=& \prod_{j\in J} w(\varepsilon_jx_j)\,
\prod_{\stackrel{j,j^\prime \in J}{j