Canonical variables of the second kind and the reduction of the N-Body Problem

We introduce a new kind of canonical variables that prove very useful when the order of a Hamiltonian system can be reduced by one, as in the case of isoenergetic reduction, and of what we call homogeneous reduction. The Kepl er Problem, Geometrical Optics and McGehee Blow-up are discussed as example s. Finally we carry out the isoenergetic reduction of the general N-Body Pr oblem using the new variables, and briefly discuss its application to the p roblem of collision.