Abstract
In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. As contribution, we obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system.
Highlights
The Morales-Ramis theory is a powerful tool for showing the nonintegrability of Hamiltonian systems
In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory
We obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system
Summary
The Morales-Ramis theory is a powerful tool for showing the nonintegrability of Hamiltonian systems. In this paper we present a short material concerning to some results in Morales-Ramis theory, which relates two different notions of integrability: Integrability of Hamiltonian systems through Liouville Arnold theorem and integrability of linear differential equations through differential Galois theory. We obtain the abelian differential Galois group of the variational equation related to a bi-parametric Hamiltonian system. To understand the Morales-Ramis theory, we need to introduce two different notions of integrability: the integrability of Hamiltonian systems in Liouville sense and the integrability of linear differential equations in Picard-Vessiot sense.
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