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Long-time averaging for integrable Hamiltonian dynamics. (English) Zbl 1084.65126

This paper is concerned with the explicit computation of the space average of an observable \( A(p,q)\) over an invariant manifold \({\mathbb M}\) in a Hamiltonian system completely integrable. Assuming a completely integrable Hamiltonian system with \(d\) degrees of freedom in the sense of Arnold-Liouville, there exist \(d\) invariants \( I_1, \ldots , I_d\) in involution and \({\mathbb M}\) is defined by \({\mathbb M}= \{ (p,q) \in {\mathbb R}^d \times {\mathbb R}^d , I_j(p,q)=I_j(p_0,q_0), j=1, \ldots d \}\) then the average of \(A\) over \( {\mathbb M}(p_0,q_0)\) can be obtained through the limit of the time average \[ \lim_{T \to \infty} T^{-1} \int_0^T A(p(t),q(t)) dt \tag{1} \] where \((p(t),q(t))\) is the solution of the Hamiltonian system corresponding to the initial conditions \((p_0,q_0)\). However since the convergence of the above integral is of order \( T^{-1}\) a direct estimation of (1) becomes a lengthy calculation and therefore the search of alternative estimations with faster convergence rates is useful from a practical point of view.
To motivate the proposed approach, the authors start with the simple case of the \(d\)-dimensional harmonic oscillator to show that this rate of convergence can be considerably improved by using some iterated averages. Then, this technique is extended to the general case by means of some filter functions. Further they consider the practical situation in which the solution is approximated by a (symplectic) Runge-Kutta method with a fixed step size deriving asymptotic estimates that take into account both the limit and the Runge-Kutta approximation. After a detailed study of the continuous, semi discrete and discrete averages, the paper ends with some numerical experiments with a perturbed plane two body problem with a perturbation similar to the case of the main satellite problem which retains both the energy and angular momentum.

MSC:

65P10 Numerical methods for Hamiltonian systems including symplectic integrators
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37M15 Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70F05 Two-body problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
70M20 Orbital mechanics
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References:

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