Summary.
Given a Hamiltonian dynamical system, we address the question of computing the limit of the time-average of an observable. For a completely integrable system, it is known that ergodicity can be characterized by a diophantine condition on its frequencies and that this limit coincides with the space-average over an invariant manifold. In this paper, we show that we can improve the rate of convergence upon using a filter function in the time-averages. We then show that this convergence persists when a symplectic numerical scheme is applied to the system, up to the order of the integrator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russian Math. Surveys 18, 85–191 (1963)
Arnold, V.I.: Mathematical methods of classical mechanics. Volume 60 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1978
Cancès, E., Castella, F., Chartier, P., Faou, E., Le Bris, C., Legoll, F., Turinici, G.: High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations. Submitted to J. Chem. Phys., 2003
Do Carmo, M.P.: Riemannian Geometry. Series Mathematics: Theory and Applications. Birkhäuser, Boston, 1992
Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 2002. Structure-preserving algorithms for ordinary differential equations
Kolmogorov, A.N.: On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98, 527–530 (1954)
Kolmogorov, A.N.: General theory of dynamical systems and classical mechanics. Proc. Int. Congr. Math. Amsterdam 1954, Vol. 1, pp. 315–333
Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. K1. 1962, pp. 1–20
Nekhoroshev, N.N.: An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russ. Math. Surveys 32, 1–65 (1977)
Papoulis, A. Signal Analysis. Electrical and Electronic Engineering Series. McGraw-Hill, Singapore, 1984
Shang, Z.: KAM theorem of symplectic algorithms for Hamiltonian systems. Numer. Math. 83, 477–496 (1999)
Shang, Z.: Resonant and Diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity 13 299–308 (2000)
Suzuki, M., Umeno, K.: Higher-order decomposition theory of exponential operators and its applications to QMC and nonlinear dynamics. In: Computer Simulation Studies in Condensed-Matter Physics VI, Landau, Mon, Schüttler (eds.), Springer Proceedings in Physics 76, 74–86 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cancès, E., Castella, F., Chartier, P. et al. Long-time averaging for integrable Hamiltonian dynamics. Numer. Math. 100, 211–232 (2005). https://doi.org/10.1007/s00211-005-0599-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0599-0