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arXiv · 2607.17407

Fouling maps and polynomial first integrals from symmetric tensor fields

Rafael Azuaje·Juan Carlos Marrero·Edith Padrón
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Abstract

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.

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Rafael Azuaje, Juan Carlos Marrero, Edith Padrón. 2026-07-19. Fouling maps and polynomial first integrals from symmetric tensor fields. https://arxiv.org/abs/2607.17407

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