Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Mather's $ \alpha$-function of mechanical systems

Author: Wei Cheng
Journal: Proc. Amer. Math. Soc. 139 (2011), 2143-2149
MSC (2010): Primary 37Jxx, 70Hxx
Published electronically: November 22, 2010
MathSciNet review: 2775392
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study Mather's $ \alpha$-function for mechanical systems. We show that for mechanical systems, the $ \alpha$-function is differentiable at $ c=0$ in at least one direction. We also give a topological condition on the potential function to guarantee the existence of a flat part near $ c=0$ for general mechanical systems. Some examples are also given.

References [Enhancements On Off] (What's this?)

  • [Ba] Bangert, V., Minimal geodesics, Ergodic Theory Dynam. Systems, 10(1990), no. 2, 263-286. MR 1062758 (91j:58126)
  • [Be] Bernard, P., Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble), 52(2002), no. 5, 1533-1568. MR 1935556 (2003m:37088)
  • [BC] Bernard, P. and Contreras, G., A generic property of families of Lagrangian systems, Ann. of Math. (2), 167(2008), no. 3, 1099-1108. MR 2415395 (2009d:37113)
  • [BIK] Burago, D., Ivanov, S. and Kleiner, B., On the structure of the stable norm of periodic metrics, Math. Res. Lett. 4(1997), no. 6, 791-808. MR 1492121 (98k:53051)
  • [C] Carneiro, M. J. Dias, On minimizing measures of the action of autonomous Lagrangians, Nonlinearity, 8(1995), no. 6, 1077-1085. MR 1363400 (96j:58062)
  • [CIPP] Contreras, G., Iturriaga, R., Paternain, G. P. and Paternain, M., Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8(1998), no. 5, 788-809. MR 1650090 (99f:58075)
  • [Fab] Farber, M., Topology of closed one-forms. Mathematical Surveys and Monographs, 108. American Mathematical Society, Providence, RI, 2004. MR 2034601 (2005c:58023)
  • [Fat] Fathi, A., Weak KAM theorem in Lagragian dynamics, to be published by Cambridge University Press.
  • [FS] Fathi, A. and Siconolfi, A., Existence of $ C^1$ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155(2004), no. 2, 363-388. MR 2031431 (2004m:37114)
  • [LPV] Lions, P. L., Papanicolaou, G. and Varadhan, S. R. S., Homogenization of Hamilton-Jacobi equations, unpublished manuscript, 1988.
  • [Man1] Mañé, R., Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9(1996), no. 2, 273-310. MR 1384478 (97d:58118)
  • [Mas1] Massart, D., On Aubry sets and Mather's action functional, Israel J. Math., 134(2003), 157-171. MR 1972178 (2004g:37088)
  • [Mas2] Massart, D., Vertices of Mather's beta function. II, Ergodic Theory Dynam. Systems, 29(2009), no. 4, 1289-1307. MR 2529650
  • [Mat1] Mather, John N., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207(1991), no. 2, 169-207. MR 1109661 (92m:58048)
  • [Mat2] Mather, John N., Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble), 43(1993), no. 5, 1349-1386. MR 1275203 (95c:58075)
  • [Mat3] Mather, John N., Examples of Aubry sets, Ergodic Theory Dynam. Systems, 24(2004), no. 5, 1667-1723. MR 2104599 (2005h:37143)
  • [O] Osuna, O., Vertices of Mather's beta function, Ergodic Theory Dynam. Systems, 25(2005), no. 3, 949-955. MR 2142954 (2005m:37150)
  • [Y] Yu, Y., Properties of effective Hamiltonian and connections with Aubry-Mather theory in two dimension, preprint, 2009.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 37Jxx, 70Hxx

Retrieve articles in all journals with MSC (2010): 37Jxx, 70Hxx

Additional Information

Wei Cheng
Affiliation: Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

Keywords: Mather theory, $\alpha$-function, mechanical systems
Received by editor(s): December 30, 2009
Received by editor(s) in revised form: June 11, 2010
Published electronically: November 22, 2010
Additional Notes: This work was partially supported by the National Basic Research Program of China (Grant No. 2007CB814800) and Natural Scientific Foundation of China (Grant No. 10971093)
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society