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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...

In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V\left(q\right)$, $q\in {ℂ}^n$, of degree $k\in {\mathbb{Z}}^{☆}$. The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V^{\prime \prime }\left(c\right)$ calculated at a non-zero point $c\in {ℂ}^n$, such that $V^{\prime }\left(c\right)=c$. The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{\prime \prime }\left(c\right)$ is not diagonalizable....

We show how using the differential Galois theory one can find effectively necessary conditions for the integrability of Hamiltonian systems with homogeneous potentials.

We report our recent results concerning integrability of Hamiltonian systems governed by Hamilton’s function of the form $H=1/2{\sum }_{i=1}^{n}p{²}_i+V\left(q\right)$, 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...