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This article is cited in 11 scientific papers (total in 11 papers)
Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom
M. Przybylska Toruń Centre for Astronomy, N. Copernicus University,
Gagarina 11, PL-87–100 Toruń, Poland
Abstract:
We consider natural complex Hamiltonian systems with $n$ degrees of
freedom given by a Hamiltonian function which is a sum of the standard
kinetic energy and a homogeneous polynomial potential $V$ of degree
$k>2$. The well known Morales–Ramis theorem gives the strongest known
necessary conditions for the Liouville integrability of such systems. It
states that for each $k$ there exists an explicitly known infinite set
${\mathcal M}_k\subset{\mathbb Q}$ such that if the system is integrable,
then all eigenvalues of the Hessian matrix $V''({\boldsymbol d})$ calculated at a
non-zero ${\boldsymbol d}\in{\mathbb C}^n$ satisfying $V'({\boldsymbol d})={\boldsymbol d}$,
belong to ${\mathcal M}_k$.
The aim of this paper is, among others, to sharpen this result. Under
certain genericity assumption concerning $V$ we prove the following fact.
For each $k$ and $n$ there exists a finite set ${\mathcal I}_{n,k}\subset{\mathcal M}_k$
such that if the system is integrable, then all eigenvalues of the Hessian
matrix $V''({\boldsymbol d})$ belong to ${\mathcal I}_{n,k}$. We give an algorithm which
allows to find sets ${\mathcal I}_{n,k}$.
We applied this results for the case $n=k=3$ and we found all integrable
potentials satisfying the genericity assumption. Among them several are
new and they are integrable in a highly non-trivial way. We found three
potentials for which the additional first integrals are of degree 4 and 6
with respect to the momenta.
Keywords:
integrability, Hamiltonian systems, homogeneous potentials, differential Galois group.
Received: 30.05.2008 Accepted: 14.01.2009
Citation:
M. Przybylska, “Darboux Points and Integrability of Homogeneous Hamiltonian Systems with Three and More Degrees of Freedom”, Regul. Chaotic Dyn., 14:2 (2009), 263–311
Linking options:
https://www.mathnet.ru/eng/rcd551 https://www.mathnet.ru/eng/rcd/v14/i2/p263
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