Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential

Maciejewski, Andrzej J.; Przybylska, Maria

doi:10.1063/1.1917311

“…The above theorem gives a very strong generalisation of Theorem 2.4 in [22] for systems with an arbitrary number of degrees of freedom.…”

Section: Is Integrable With Rational First Integrals Then Matrix V ′mentioning

confidence: 60%

“…To this end we proceed as in [22]. Namely, let PO(n, C) be the complex projective orthogonal subgroup of GL(n, C), i.e., PO(n, C) = {A ∈ GL(n, C),…”

Section: Remark 12 As It Was Explained In [10] the Assumption That Vmentioning

confidence: 99%

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

Przybylska

¹

2009

View full text Add to dashboard Cite

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 M k ⊂ Q such that if the system is integrable, then all eigenvalues of the Hessian matrixThe 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 I n,k ⊂ M k such that if the system is integrable, then all eigenvalues of the Hessian matrix V ′′ (d) belong to I n,k . We give an algorithm which allows to find sets 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 nontrivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta. MSC2000 numbers: 37J30, 70H07, 37J35, 34M35.

show abstract

“…Let us note that this lemma is a generalisation of Corollary 2.1 in [22], when the case n = 2 was considered and then a generic potential has D(2, k) = k proper Darboux points. …”

Section: Lemma 22 the Set Of Generic Potentialsmentioning

confidence: 65%

“…Thus, in an appropriate base, the variational equations along solution (2.41) contain equationη = −λt k−2 η. As it was shown in [22] the differential Galois group of this equation is SL(2, C) and hence the identity component of the differential Galois group of all variational equations is not Abelian. Thus, by Morales-Ramis Theorem 1.1, the system is not integrable.…”

Section: Is Integrable With Rational First Integrals Then Matrix V ′mentioning

confidence: 84%

See 1 more Smart Citation

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

Przybylska

¹

2009

View full text Add to dashboard Cite

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 M k ⊂ Q such that if the system is integrable, then all eigenvalues of the Hessian matrixThe 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 I n,k ⊂ M k such that if the system is integrable, then all eigenvalues of the Hessian matrix V ′′ (d) belong to I n,k . We give an algorithm which allows to find sets 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 nontrivial way. We found three potentials for which the additional first integrals are of degree 4 and 6 with respect to the momenta. MSC2000 numbers: 37J30, 70H07, 37J35, 34M35.

show abstract

Straight line orbits in Hamiltonian flows

Howard

¹

,

Meiss

²

2009

Celest Mech Dyn Astr

View full text Add to dashboard Cite

We investigate periodic straight-line orbits (SLO) in Hamiltonian force fields using both direct and inverse methods. A general theorem is proven for natural Hamiltonians quadratic in the momenta in arbitrary dimension and specialized to two and three dimension. Next we specialize to homogeneous potentials and their superpositions, including the familiar Hénon-Heiles problem. It is shown that SLO's can exist for arbitrary finite superpositions of Nforms. The results are applied to a family of generalized Hénon-Heiles potentials having discrete rotational symmetry. SLO's are also found for superpositions of these potentials.

show abstract

Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential

Cited by 55 publications

References 38 publications

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom

Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom. Nongeneric cases

Straight line orbits in Hamiltonian flows

Contact Info

Product

Resources

About