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Non-integrability of certain Hamiltonian systems. Applications of the Morales-Ramis differential Galois extension of Ziglin theory

100%
Maciejewski A.
Banach Center Publications
|
2002
|
tom 58
|
nr 1
139-150
EN
The aim of this paper is to present two examples of non academic Hamiltonian systems for which the Morales-Ramis theory can be applied effectively. First, we investigate the Gross-Neveu system with n degrees of freedom. Till now it has been proved that this system is not integrable for n = 3. We give a simple proof that it is not completely integrable for an arbitrary n ≥ 3. Our second example is a natural generalisation of the Jacobi problem of a material point moving on an ellipsoid. We formulate sufficient conditions for its non-integrability.
2
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Overview of the differential Galois integrability conditions for non-homogeneous potentials

63%
Maciejewski A., Przybylska M.
Banach Center Publications
|
2011
|
tom 94
|
nr 1
221-232
EN
We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton's function of the form $H = 1/2 ∑_{i=1}^{n} p²_{i} + V(q)$, where the potential V is a finite sum of homogeneous components. In this paper we show how to find, in the differential Galois framework, computable necessary conditions for the integrability of such systems. Our main result concerns potentials of the form $V = V_k + V_K$, where $V_k$ and $V_K$ are homogeneous functions of integer degrees k and K > k, respectively. We present examples of integrable systems which were obtained by applying our main theorem.
3
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Taylor obstruction to the integrability of homogeneous potentials of degree two. An application of higher order variational equations

63%
Duval G., Maciejewski A.
Banach Center Publications
|
2011
|
tom 94
|
nr 1
173-185
EN
We show how using the differential Galois theory one can find effectively necessary conditions for the integrability of Hamiltonian systems with homogeneous potentials.
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