Geodesic
First Integrals of Extended Hamiltonians in $n+1$ Dimensions Generated by Powers of an Operator
Symmetry, integrability and geometry: methods and applications, Tome 7 (2011) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: superintegrable Hamiltonian systems; polynomial first integrals; constant curvature; Hessian tensor. 
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     author = {Claudia Chanu and Luca Degiovanni and Giovanni Rastelli},
     title = {First {Integrals} of {Extended} {Hamiltonians} in $n+1$ {Dimensions} {Generated} by {Powers} of an {Operator}},
     journal = {Symmetry, integrability and geometry: methods and applications},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a37/}
}
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Claudia Chanu; Luca Degiovanni; Giovanni Rastelli. First Integrals of Extended Hamiltonians in $n+1$ Dimensions Generated by Powers of an Operator. Symmetry, integrability and geometry: methods and applications, Tome 7 (2011). http://geodesic.mathdoc.fr/item/SIGMA_2011_7_a37/

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