  Conservative symmetric second-order one-step schemes are derived for
dynamical systems describing various many-body systems using the Discrete
Multiplier Method. This includes conservative schemes for the $n$-species
Lotka-Volterra system, the $n$-body problem with radially symmetric potential
and the $n$-point vortex models in the plane and on the sphere. In particular,
we recover Greenspan-Labudde's conservative schemes for the $n$-body problem.
Numerical experiments are shown verifying the conservative property of the
schemes and second-order accuracy.
