On Mather's -function of mechanical systems

Author:
Wei Cheng

Journal:
Proc. Amer. Math. Soc. **139** (2011), 2143-2149

MSC (2010):
Primary 37Jxx, 70Hxx

DOI:
https://doi.org/10.1090/S0002-9939-2010-10643-3

Published electronically:
November 22, 2010

MathSciNet review:
2775392

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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

**[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 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P. L., Papanicolaou, G. and Varadhan, S. R. S.,*Homogenization of Hamilton-Jacobi equations*, unpublished manuscript, 1988.**[Man1]**Mañé, R.,*Generic properties and problems of minimizing measures of Lagrangian systems*, Nonlinearity,**9**(1996), no. 2, 273-310.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.

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

Email:
chengwei@nju.edu.cn

DOI:
https://doi.org/10.1090/S0002-9939-2010-10643-3

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.