First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

Details First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator First Integrals of Extended Hamiltonians in n+1 Dimensions Generated by Powers of an Operator

2011-01-31

arxiv.org/abs/1101.5975v3

SIGMA 7 (2011), 038, 12 pages

Physics

Mathematical Physics

Theorem 1, Lemmas 1 and 2, Example 2 are corrected

Scientific paper

10.3842/SIGMA.2011.038

We describe a procedure to construct polynomial in the momenta first integrals of arbitrarily high degree for natural Hamiltonians $H$ obtained as one-dimensional extensions of natural (geodesic) $n$-dimensional Hamiltonians $L$. The Liouville integrability of $L$ implies the (minimal) superintegrability of $H$. We prove that, as a consequence of natural integrability conditions, it is necessary for the construction that the curvature of the metric tensor associated with $L$ is constant. As examples, the procedure is applied to one-dimensional $L$, including and improving earlier results, and to two and three-dimensional $L$, providing new superintegrable systems.

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