Fouling maps and polynomial first integrals from symmetric tensor fields
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 notion of fouling map as a (not necessarily 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, on the 2-sphere and on a Riemannian manifold with a Liouville metric illustrate the method.
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BibTeX
@misc{azuaje2026fouling,
title = {{Fouling maps and polynomial first integrals from symmetric tensor fields}},
author = {Rafael Azuaje and Juan Carlos Marrero and Edith Padrón},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.17407},
eprint = {2607.17407},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.17407},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fouling maps and polynomial first integrals from symmetric tensor fields
AU - Rafael Azuaje
AU - Juan Carlos Marrero
AU - Edith Padrón
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.17407
UR - https://arxiv.org/abs/2607.17407
AB - 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 notion of fouling map as a (not necessarily 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, on the 2-sphere and on a Riemannian manifold with a Liouville metric illustrate the method.
ER -