Numerical integration in Celestial Mechanics: a case for contact geometry

Abstract

Several dynamical systems of interest in celestial mechanics can be written in the formqFor instance, the modified Kepler problem, the spin-orbit model and the Lane-Emden equation all belong to this class. In this work we start an investigation of these models from the point of view of contact geometry. In particular we focus on the (contact) Hamiltonisation of these models and on the construction of the corresponding geometric integrators.

Geometric Numerical Integration of Liénard Systems via a Contact Hamiltonian Approach

“…For s = 0 and setting the appropriate initial condition p 0 = − f (q 0 ) − g(q 0 )/s 0 , p(t) derived from (18) turns out to be the slope of the tangent ds dq to the orbit of the system at each point (q(t), s(t)) of its evolution. This stems from the fact that (16)- (18) are the characteristic equations of the Hamilton-Jacobi equation for (15). Details of this derivation are in preparation by [28].…”

“…Contact splitting integrators are a class of geometric integrators recently introduced in the context of celestial mechanics [15]. They are the contact analogues of the well-known symplectic splitting integrators.…”

“…One of the main advantages of using contact splitting integrators is the possibility to have a direct error control by using the modified equations obtained from the modified Hamiltonian that results from the Baker-Campbell-Hausdorff (BCH) formula (see [15] for further details on the derivation of the modified Hamiltonian in the contact case). Indeed, for an integrator of order 2n multiple applications of the BCH formula give [15]…”

“…Contact geometry was introduced in Sophus Lie's study of differential equations, and has been the subject of intense research, especially related to low-dimensional topology [8]. In recent years, contact Hamiltonian systems have found many applications, first in the context of thermodynamics [9][10][11] and, more recently, in the context of the Hamiltonisation of several dissipative dynamical systems [12][13][14][15][16][17][18][19]. The large number of applications of contact systems that have appeared recently motivated research on geometric numerical integration [15,16,20,21].…”

“…In recent years, contact Hamiltonian systems have found many applications, first in the context of thermodynamics [9][10][11] and, more recently, in the context of the Hamiltonisation of several dissipative dynamical systems [12][13][14][15][16][17][18][19]. The large number of applications of contact systems that have appeared recently motivated research on geometric numerical integration [15,16,20,21]. Fortunately, contact flows possess geometric integrators (both variational and Hamiltonian) that precisely parallel their symplectic counterparts, and therefore they show remarkable numerical and analytical properties such as, e.g., increased stability, near-preservation of invariant quantities, and modified Hamiltonians.…”