Abstract
We give a rigorous proof of the equivalence of Mañé’s supercritical potential and the minimal action with respect to an associated Jacobi-Finsler metric. As a consequence, we give an explicit representation of the weak KAM solutions of one-dimensional mechanical systems without the quadratic assumption on the kinetic energy term of the Hamiltonians, and a criterion of the integrability result for such a system of arbitrary degree of freedom by the regularity assumption on Mather’s α-function is discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Mather, J. N.: Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z., 207, 169–207 (1991)
Mather, J. N.: Variational construction of connecting orbits. Ann. Inst. Fourier (Grenoble), 43, 1349–1386 (1993)
Fathi, A., Siconolfi, A.: PDE aspects of Aubry-Mather theory for quasiconvex Hamiltonians. Calc. Var. Partial Differential Equations, 22, 185–228 (2005)
Arnold, V. I.: Mathematical Methods of Classical Mechanics, 2nd ed., Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989
Contreras, G., Iturriaga, R., Paternain, G. P., et al.: Lagrangian graphs, minimizing measures and Mañé’s critical values. Geom. Funct. Anal., 8, 788–809 (1998)
Iturriaga, R., Sánchez-Morgado, H.: Finsler metrics and action potentials. Proc. Amer. Math. Soc., 128, 3311–3316 (2000)
Evans, L. C., Gomes, D.: Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch. Ration. Mech. Anal., 157, 1–33 (2001)
Fathi, A.: Weak KAM Theorem in Lagragian Dynamics, to be published by Cambridge University Press
Fathi, A., Siconolfi, A.: Existence of C 1 critical subsolutions of the Hamilton-Jacobi equation. Invent. Math., 155, 363–388 (2004)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis, Grundlehren Text Editions, Springer-Verlag, Berlin, 2001
Burago, D., Ivanov, S., Kleiner, B.: On the structure of the stable norm of periodic metrics. Math. Res. Lett., 4, 791–808 (1997)
Lions, P. L., Papanicolaou, G., Varadhan, S. R. S.: Homogenization of Hamilton-Jacobi equations. Unpublished manuscript, 1988
Cheng, W.: The integrability of positively definite Lagrangian systems via variational criterion: mechanical systems. J. Differential Equations, 249, 1664–1673 (2010)
Cheng, W.: On the Mather’s α-function of mechanical systems. Proc. Amer. Math. Soc., 139, 2143–2149 (2011)
Bangert, V.: Minimal geodesics. Ergodic Theory Dynam. Systems, 10, 263–286 (1990)
Burago, D., Ivanov, S.: Riemannian tori without conjugate points are flat. Geom. Funct. Anal., 4, 259–269 (1994)
Mañé, R.: Generic properties and problems of minimizing measures of Lagrangian systems. Nonlinearity, 9, 273–310 (1996)
Mañé, R.: Lagrangian flows: the dynamics of globally minimizing orbits. Bol. Soc. Brasil. Mat. (N.S.), 28, 141–153 (1997)
Mather, J. N.: Examples of Aubry sets. Ergodic Theory Dynam. Systems, 24, 1667–1723 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Basic Research Program of China (Grant No. 2007CB814800) and Natural Scientific Foundation of China (Grant No. 10971093)
Electronic supplementary material
Rights and permissions
About this article
Cite this article
Cheng, W. Generalized Maupertuis’ principle with applications. Acta. Math. Sin.-English Ser. 28, 2153–2160 (2012). https://doi.org/10.1007/s10114-012-1001-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-012-1001-7