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TitleA reduced form for linear differential systems and its application to integrability of Hamiltonian systems
Author(s) Ainhoa Aparicio-Monforte, Jacques-Arthur Weil
TypeArticle in Journal
AbstractLet k be a differential field with algebraic closure k ̄ , and let [ A ] : Y ′ = A Y with A ∈ M n ( k ) be a linear differential system. Denote by g the Lie algebra of the differential Galois group of [ A ] . We say that a matrix R ∈ M n ( k ¯ ) is a reduced form of [ A ] if R ∈ g ( k ¯ ) and there exists P ∈ G L n ( k ¯ ) such that R = P − 1 ( A P − P ′ ) ∈ g ( k ¯ ) . Such a form is often the sparsest possible attainable through gauge transformations without introducing new transcendents. In this paper, we discuss how to compute reduced forms of some symplectic differential systems, arising as variational equations of Hamiltonian systems. We use this to give an effective form of the Morales–Ramis theorem on (non-)-integrability of Hamiltonian systems.
KeywordsDifferential Galois theory, Computer algebra, Hamiltonian Systems, Integrability, Morales–Ramis theory, Reduced form
URL http://www.sciencedirect.com/science/article/pii/S0747717111001465
JournalJournal of Symbolic Computation
Pages192 - 213
Translation No
Refereed No