Separability and Lax pairs for Hénon–Heiles system

Abstract

Scattering theory is a convenient way to describe systems that are subject to time-dependent perturbations which are localized in time. Using scattering theory, one can compute time-dependent invariant objects for the perturbed system knowing the invariant objects of the unperturbed system. In this paper, we use scattering theory to give numerical computations of invariant manifolds appearing in laser-driven reactions. In this setting, invariant manifolds separate regions of phase space that lead to different outcomes of the reaction and can be used to compute reaction rates. V C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4767656] In this paper, we implement numerically the theoretical ideas of Blazevski and de la Llave (2011) using the laser-driven H enon-Heiles system introduced in Kawai et al. (2007). The ideas of this paper are robust and can be used for any system subject to a time-dependent perturbation localized in time. Using the intertwining relations, scattering theory yields a time-dependent conjugacy between the perturbed and unperturbed systems. Assuming one has the relevant invariant manifolds for the unperturbed system, we exploit the conjugacy to compute the corresponding objects for the perturbed system. The algorithm used in this paper to compute invariant manifolds is an alternative to the use of time-dependent normal form theory used in Kawai et al. (2007) and is readily parallelizable.

On the Chaplygin system on the sphere with velocity dependent potential

“…Similar to [15] determinant of this Lax matrix is a product of the separated relations Φ + Φ − . We can add to this product Φ + Φ − (4.15) some termŝ…”

“…Following [15] we can use the separated relations (4.15) in order to get trivial 4 × 4 Lax matrix for this system…”

Untitled

“…For µ = 0 from (36)-(37) and (26), one easily recovers the known solution [21] expressed with Weierstrass elliptic functions:…”

Integration of a generalized Hénon–Heiles Hamiltonian

“…In both cases, in the limit µ → 0, we recover the previous results. 9,10 In section VI, we compare our method of integration, starting from the Hamiltonian system, with the method used by Cosgrove 13 for integrating the fourth order ODE's (11) and (13).…”

“…For µ = 0, the equations of motion for the SK and KK cases have been integrated [8][9][10] in terms of elliptic functions.…”