Classical Integrability of the Calogero-Moser Systems

Abstract

We have seen that if one defines an N × N matrix L as

$$ \left( L \right)_{jk} = \delta _{jk} p_j + ig\frac{{\left( {1 - \delta _{jk} } \right)}} {{q_j - q_k }} $$

(2.1)

one can write the equations of motion belonging to

$$ H = \frac{1} {2}\left( {\sum\limits_{i = 1}^N {p_i^2 } + g^2 \sum\limits_{i \ne 1}^N {\frac{1} {{(q_i - q_j )^2 }}} } \right) $$

(2.2)

in the form

$$ \dot L = \left[ {L,M} \right] , $$

(2.3)

where M is given by

$$ M_{jk} = ig\delta _{jk} \sum\limits_{l \ne j} {\frac{1} {{(q_j - q_l )^2 }} - ig\frac{{(1 - \delta _{jk} )}} {{(q_j - q_k )^2 }}} . $$

(2.4)