Contact canonoid maps and their conserved and dissipated quantities

Abstract

We study conserved and dissipated quantities associated with contact canonoid transformations and their extension to possibly singular smooth maps. The defining condition is imposed on the pulled-back contact one-form, without requiring it to remain contact. A polynomial construction produces first integrals from one-forms having the same dissipation rate as the original contact form. Two normalizations extend this construction to maps whose associated Hamiltonians have different dissipation rates. On the set where the associated Hamiltonian is nonzero, we construct a smooth invariant endomorphism whose traces are conserved. This construction includes zeros of the original Hamiltonian, where the endomorphism vanishes. We relate the tensor traces to the coefficients of the Hamiltonian-normalized polynomial and obtain an upper bound of n functionally independent trace invariants in dimension 2n + 1. A five-dimensional singular map attains this bound. Examples include singular maps for the damped free particle, a transformation with unequal dissipation rates, the damped harmonic oscillator, and motion under gravity with linear friction.