2026/07/19 by Rafael Azuaje, Juan Carlos Marrero, Edith Padrón
#math-ph #math.MP
Under the framework of time-independent Hamiltonian mechanics on the cotangent bundles T^*Q of the configuration spaces Q of mechanical systems, we introduce the concept of fouling map as a non-invertible generalization of the so-called fouling transformations --canonoid transformations preserving configuration coordinates--. We develop a tensorial method for constructing polynomial fouling maps. We show that each such map induces a (1,1)-tensor field invariant under the Hamiltonian flow, whose traces of its powers are polynomial constants of motion. For mechanical Hamiltonian functions --the kinetic energy plus the potential energy on a semi-Riemannian configuration space (Q,g)--, we completely characterize polynomial bundle maps arising from symmetric (k+1,0)-tensor fields and derive the conditions ensuring their fouling nature. Several explicit examples on the Euclidean plane and on the 2-sphere illustrate the method.