The scattering matrix and associated formulas in Hamiltonian mechanics

TL;DR

This paper explores scattering theory within Hamiltonian mechanics, drawing parallels with quantum scattering, and establishes identities linking the scattering symplectomorphism's invariants to physical quantities like time delay.

Contribution

It introduces the scattering symplectomorphism in Hamiltonian mechanics and proves identities relating its invariants to phase space volume and time delay, analogous to quantum formulas.

Findings

Relation between Calabi invariant and total time delay

Identity linking scattering symplectomorphism to phase space volume

Analogies with Birman-Krein and Eisenbud-Wigner formulas

Abstract

We survey the basic notions of scattering theory in Hamiltonian mechanics with a particular attention to the analogies with scattering theory in quantum mechanics. We discuss the scattering symplectomorphism, which is analogous to the scattering matrix. We prove identities which relate the Calabi invariant of the scattering symplectomorphism to the total time delay and the regularised phase space volume. These identities are analogous to the Birman-Krein formula and the Eisenbud-Wigner formula in quantum scattering theory.