Integrability of potentials of degree \(k \neq \pm 2\). Second order variational equations between Kolchin solvability and abelianity. (English) Zbl 1333.37077
The paper studies the integrability of Hamiltonian systems
\[
H(q,p) = \frac{1}{2} \sum_{i=1}^n p_i^2 + V(q), \quad q,p \in \mathbb C^n
\]
where \(V(q)\) is a homogeneous function of degree \(k \in \mathbb Z \setminus \{0\}\) with \(|k| \geq 3\). In particular, it investigates second-order variational equations along a particular solution given by \(q_0(t) = \varphi(t)d\), \(\tfrac12 \dot\varphi(t) +k\varphi(t)^k=e\), where \(d \in \mathbb C^n \setminus \{0\}\) is a solution of \(V'(d)=d\), and \(e\) is a constant of energy. The paper’s major results give strong necessary conditions guaranteeing that the Galois groups of specific associated systems are virtually abelian and thus necessary conditions for integrability. The formulated conditions give effective algorithms to test such possibilities in particular cases.
Reviewer: Konstantinos Efstathiou (Suzhou)
MSC:
37J30 | Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria) |
70H07 | Nonintegrable systems for problems in Hamiltonian and Lagrangian mechanics |
37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
Keywords:
integrability; Hamiltonian systems; homogeneous potentials; differential Galois group; higher order variational equationsReferences:
[1] | E. G. C. Poole, <em>Introduction to the Theory of Linear Differential Equations</em>,, Dover Publications Inc. (1960) · Zbl 0090.30202 |
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