A simple global representation for second-order normal forms of Hamiltonian systems relative to periodic flows

Abstract

Abstract. We study the determination of the second-order normal form for perturbed Hamiltonians2 H 2 , relative to the periodic flow of the unperturbed Hamiltonian H 0 . The formalism presented here is global, and can be easily implemented in any CAS. We illustrate it by means of two examples: the Hénon-Heiles and the elastic pendulum Hamiltonians.

Hamiltonian Dynamical Systems: Symbolical, Numerical and Graphical Study

“…Suppose now that X is the generator of an S 1 −action, so the flow Fl t X is periodic in the variable t. This property can be used to put H in normal form (for details, see [1]). To this end, it is essential to define two averaging operators acting on observables.…”

“…In the particular case of a perturbed Hamiltonian, of the form H = H 0 + εH 1 + ε 2 2 H 2 + · · · , if the non-perturbed part H 0 generates an S 1 −action in such a way that its flow is periodic with frequency function w, it can be proved (see [1]) that its second-order normal form is…”

A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator

“…As previously mentioned, these equations are usually solved by writing everything in action-angle variables, using some Fourier analysis and then averaging over angles on orbits with constant action. The idea in [1] was to solve the homological equations in a global setting, again using averaging operators, but this time constructing them by means of geometric properties of the flow of Hamiltonian vector fields, thus avoiding action-angle variables and the requirement that M be symplectic. In what follows we offer a brief summary of the results in [1], in particular the explicit expressions for the normal forms to first and second order.…”

“…where the relation between the variable u and the set (q 1 , q 2 , p 1 , p 2 ) is obtained through a series of substitutions and hyperbolic rotations that will not be needed here (they come from the application of Ostrogadski's second-order formalism, see [14] and [17]). Due to the presence of a periodic flow, we will find it convenient to use the techniques in [1], considering Λ as a perturbation parameter (in fact, here Λ is a small parameter, see [17]). We begin by noticing that the Hamiltonian flow Fl t X H 0 is given by…”

G-Invariant Deformations of Almost-Coupling Poisson Structures

“…These results can be used to construct normal forms for perturbed dynamics associated to almost-coupling deformations of foliated Poisson manifolds with symmetry. Typically, such perturbed dynamics appear in the context of adiabatic theory [1] on nontrivial phase spaces, particularly those in which no global action-angle coordinates can be introduced [2,3,4].…”