The Scattering Matrix and Associated Formulas in Hamiltonian Mechanics

We prove two new identities in scattering theory in Hamiltonian mechanics and discuss the analogy between these identities and their counterparts in quantum scattering theory. These identities involve the Poincare scattering map, which is analogous to the scattering matrix. The first of our identities states that the Calabi invariant of the Poincare scattering map can be expressed as the regularised phase space volume. This is analogous to the Birman-Krein formula. The second identity relates the Poincare scattering map to the total time delay and is analogous to the Eisenbud-Wigner formula.