Phase portraits of separable Hamiltonian systems

descriptionPublicationkeyboard_double_arrow_right Article , Conference object 01 Aug 2011 Spain English Publisher:Elsevier BVJournal:Nonlinear Analysis: Theory, Methods & Applications, volume 74, pages 4,012-4,035 (issn: 0362-546X,

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Authors: Guillamon Grabolosa, Antoni; Pantazi, Chara;

doi: 10.1016/j.na.2011.03.030

handle: 11441/35792 , 2117/2258

Phase portraits of separable Hamiltonian systems

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Abstract

We study some generalizations of potential Hamiltonian systems $(H(x, y) = y^2 + F(x))$ with one degree of freedom. In particular, we are interested in Hamiltonian systems with Hamiltonian functions of type $H(x, y) = F(x) + G(y)$ arising in applied mechanical problems. We present an algorithm to plot the phase portrait (include the behavior at infinity) of any Hamiltonian system of type $H(x, y) = F(x)+G(y)$, where $F$ and $G$ are arbitrary polynomials. We are able to give the full description in the Poincaré disk according to the graphs of $F$ and $G$, extending the well-known method for the “finite” phase portrait of potential systems. Peer Reviewed

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Keywords

Differential equations, ergodic theory, :34 Ordinary differential equations::34C Qualitative theory [Classificació AMS], bifurcation diagram, Classificació AMS::34 Ordinary differential equations::34C Qualitative theory, Nonlinear Dynamics., Nonlinear oscillations and coupled oscillators for ordinary differential equations, differential equations, :37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems [Classificació AMS], phase portrait, Sistemes dinàmics diferenciables, mechanic of particles, Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory, Partícules (Física nuclear), separable Hamiltonian system, Classificació AMS::70 Mechanics of particles and systems::70K Nonlinear dynamics, Hamilton, Sistemes de, :37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory [Classificació AMS], Differentiable dynamical systems, Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems, Equacions diferencials ordinàries, Hamiltonian systems, Symmetries, invariants of ordinary differential equations, Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems, :70 Mechanics of particles and systems::70K Nonlinear dynamics [Classificació AMS]

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