On Mather’s 𝛼-function of mechanical systems

Cheng, Wei

doi:10.1090/S0002-9939-2010-10643-3

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
DOI: https://doi.org/10.1090/S0002-9939-2010-10643-3
Published electronically: November 22, 2010
MathSciNet review: 2775392
Full-text PDF Free Access

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 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] Victor Bangert, Minimal geodesics, Ergodic Theory Dynam. Systems 10 (1990), no. 2, 263–286. MR 1062758, https://doi.org/10.1017/S014338570000554X
[Be] Patrick Bernard, Connecting orbits of time dependent Lagrangian systems, Ann. Inst. Fourier (Grenoble) 52 (2002), no. 5, 1533–1568 (English, with English and French summaries). MR 1935556
[BC] Patrick Bernard and Gonzalo Contreras, A generic property of families of Lagrangian systems, Ann. of Math. (2) 167 (2008), no. 3, 1099–1108. MR 2415395, https://doi.org/10.4007/annals.2008.167.1099
[BIK] D. Burago, S. Ivanov, and B. Kleiner, On the structure of the stable norm of periodic metrics, Math. Res. Lett. 4 (1997), no. 6, 791–808. MR 1492121, https://doi.org/10.4310/MRL.1997.v4.n6.a2
[C] M. J. Dias Carneiro, On minimizing measures of the action of autonomous Lagrangians, Nonlinearity 8 (1995), no. 6, 1077–1085. MR 1363400
[CIPP] G. Contreras, R. Iturriaga, G. P. Paternain, and M. Paternain, Lagrangian graphs, minimizing measures and Mañé’s critical values, Geom. Funct. Anal. 8 (1998), no. 5, 788–809. MR 1650090, https://doi.org/10.1007/s000390050074
[Fab] Michael Farber, Topology of closed one-forms, Mathematical Surveys and Monographs, vol. 108, American Mathematical Society, Providence, RI, 2004. MR 2034601
[Fat] Fathi, A., Weak KAM theorem in Lagragian dynamics, to be published by Cambridge University Press.
[FS] Albert Fathi and Antonio Siconolfi, Existence of 𝐶¹ critical subsolutions of the Hamilton-Jacobi equation, Invent. Math. 155 (2004), no. 2, 363–388. MR 2031431, https://doi.org/10.1007/s00222-003-0323-6
[LPV] Lions, P. L., Papanicolaou, G. and Varadhan, S. R. S., Homogenization of Hamilton-Jacobi equations, unpublished manuscript, 1988.
[Man1] Ricardo Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 (1996), no. 2, 273–310. MR 1384478, https://doi.org/10.1088/0951-7715/9/2/002
[Mas1] Daniel Massart, On Aubry sets and Mather’s action functional, Israel J. Math. 134 (2003), 157–171. MR 1972178, https://doi.org/10.1007/BF02787406
[Mas2] Daniel Massart, Vertices of Mather’s beta function. II, Ergodic Theory Dynam. Systems 29 (2009), no. 4, 1289–1307. MR 2529650, https://doi.org/10.1017/S0143385708000631
[Mat1] John N. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z. 207 (1991), no. 2, 169–207. MR 1109661, https://doi.org/10.1007/BF02571383
[Mat2] John N. Mather, Variational construction of connecting orbits, Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349–1386 (English, with English and French summaries). MR 1275203
[Mat3] John N. Mather, Examples of Aubry sets, Ergodic Theory Dynam. Systems 24 (2004), no. 5, 1667–1723. MR 2104599, https://doi.org/10.1017/S0143385704000446
[O] Osvaldo Osuna, Vertices of Mather’s beta function, Ergodic Theory Dynam. Systems 25 (2005), no. 3, 949–955. MR 2142954, https://doi.org/10.1017/S0143385704000835
[Y] Yu, Y., Properties of effective Hamiltonian and connections with Aubry-Mather theory in two dimension, preprint, 2009.