April  2020, 13(4): 1395-1410. doi: 10.3934/dcdss.2020078

Effective Hamiltonian dynamics via the Maupertuis principle

1. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

2. 

3 Badgers Hollow, Peparharow Road, Godalming GU7 2PX, United Kingdom

* Corresponding author: Johannes Zimmer

Received  December 2017 Revised  September 2018 Published  April 2019

We consider the dynamics of a Hamiltonian particle forced by a rapidly oscillating potential in $ m $-dimensional space. As alternative to the established approach of averaging Hamiltonian dynamics by reformulating the system as Hamilton-Jacobi equation, we propose an averaging technique via reformulation using the Maupertuis principle. We analyse the result of these two approaches for one space dimension. For the initial value problem the solutions converge uniformly when the total energy is fixed. If the initial velocity is fixed independently of the microscopic scale, then the limit solution depends on the choice of subsequence. We show similar results hold for the one-dimensional boundary value problem. In the higher dimensional case we show a novel connection between the Hamilton-Jacobi and Maupertuis approaches, namely that the sets of minimisers and saddle points coincide for these functionals.

Citation: Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. Effective Hamiltonian dynamics via the Maupertuis principle. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1395-1410. doi: 10.3934/dcdss.2020078
References:
[1]

V. I. Arnol' d, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, [1989], Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition.

[2]

M. Biesiada, The power of the Maupertuis-Jacobi principle-dreams and reality, Chaos Solitons Fractals, 5 (1995), 869-879.  doi: 10.1016/0960-0779(94)E0082-Z.

[3]

A. Braides, Almost periodic methods in the theory of homogenization, Appl. Anal., 47 (1992), 259-277.  doi: 10.1080/00036819208840144.

[4]

A. Braides, Γ-convergence for Beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[5]

A. BraidesG. Buttazzo and I. Fragalà, Riemannian approximation of Finsler metrics, Asymptot. Anal., 31 (2002), 177-187. 

[6]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, vol. 12 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.

[7]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[8]

W. E, A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759.  doi: 10.1002/cpa.3160440702.

[9]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157 (2001), 1-33.  doi: 10.1007/PL00004236.

[10]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. Ⅱ, Arch. Ration. Mech. Anal., 161 (2002), 271-305.  doi: 10.1007/s002050100181.

[11]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

[12]

D. A. Gomes, Hamilton-Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, Master's thesis, University of California at Berkeley, 2000.

[13]

J. Jost, Postmodern Analysis, 3rd edition, Universitext, Springer-Verlag, Berlin, 2005.

[14]

P. E. B. Jourdain, Maupertuis and the principle of least action, Monist, 22 (1912), 414–459, URL http://www.jstor.org/stable/27900387. doi: 10.5840/monist191222331.

[15]

P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, 1987, Preprint.

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, A basic exposition of classical mechanical systems. doi: 10.1007/978-0-387-21792-5.

[17]

D. C. Sutton, Microscopic Hamiltonian Systems and their Effective Description, PhD thesis, Department of Mathematical Sciences, University of Bath, 2013.

show all references

References:
[1]

V. I. Arnol' d, Mathematical Methods of Classical Mechanics, vol. 60 of Graduate Texts in Mathematics, Springer-Verlag, New York, [1989], Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition.

[2]

M. Biesiada, The power of the Maupertuis-Jacobi principle-dreams and reality, Chaos Solitons Fractals, 5 (1995), 869-879.  doi: 10.1016/0960-0779(94)E0082-Z.

[3]

A. Braides, Almost periodic methods in the theory of homogenization, Appl. Anal., 47 (1992), 259-277.  doi: 10.1080/00036819208840144.

[4]

A. Braides, Γ-convergence for Beginners, vol. 22 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.

[5]

A. BraidesG. Buttazzo and I. Fragalà, Riemannian approximation of Finsler metrics, Asymptot. Anal., 31 (2002), 177-187. 

[6]

A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, vol. 12 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998.

[7]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Progress in Nonlinear Differential Equations and their Applications, 8, Birkhäuser Boston Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.

[8]

W. E, A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759.  doi: 10.1002/cpa.3160440702.

[9]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. I, Arch. Ration. Mech. Anal., 157 (2001), 1-33.  doi: 10.1007/PL00004236.

[10]

L. C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics. Ⅱ, Arch. Ration. Mech. Anal., 161 (2002), 271-305.  doi: 10.1007/s002050100181.

[11]

L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.

[12]

D. A. Gomes, Hamilton-Jacobi Equations, Viscosity Solutions and Asymptotics of Hamiltonian Systems, Master's thesis, University of California at Berkeley, 2000.

[13]

J. Jost, Postmodern Analysis, 3rd edition, Universitext, Springer-Verlag, Berlin, 2005.

[14]

P. E. B. Jourdain, Maupertuis and the principle of least action, Monist, 22 (1912), 414–459, URL http://www.jstor.org/stable/27900387. doi: 10.5840/monist191222331.

[15]

P.-L. Lions, G. Papanicolaou and S. R. S. Varadhan, Homogenization of Hamilton-Jacobi equations, 1987, Preprint.

[16]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, vol. 17 of Texts in Applied Mathematics, 2nd edition, Springer-Verlag, New York, 1999, A basic exposition of classical mechanical systems. doi: 10.1007/978-0-387-21792-5.

[17]

D. C. Sutton, Microscopic Hamiltonian Systems and their Effective Description, PhD thesis, Department of Mathematical Sciences, University of Bath, 2013.

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