JournalscmhVol. 72, No. 3pp. 481–499

Critical values of autonomous Lagrangian systems

  • G. Paternain

    Universidad de la República, Montevideo, Uruguay
  • Miguel Paternain

    Universidad de la República, Montevideo, Uruguay
Critical values of autonomous Lagrangian systems cover
Download PDF

Abstract

Let M be a closed manifold and L:TMRL : TM \rightarrow \bf {R} a convex superlinear Lagrangian. We consider critical values of Lagrangians as defined by R. Mañé in [5]. Let cu(L) denote the critical value of the lift of L to the universal covering of M and let ca(L) denote the critical value of the lift of L to the abelian covering of M. It is easy to see that in general, cu(L)ca(L)c_{u}(L) \leq c_{a}(L) . Let c0(L) denote the strict critical value of L defined as the smallest critical value of LωL - \omega where ω\omega ranges among all possible closed 1-forms. We show that ca(L) = c0(L). We also show that if there exists k such that the Euler-Lagrange flow of L on the energy level k' is Anosov for all kkk'\geq k , then k>cu(L)k > c_{u}(L) . Afterwards, we exhibit a Lagrangian on a compact surface of genus two which possesses Anosov energy levels with energy k<ca(L)k < c_{a}(L) , thus answering in the negative a question raised by Mañé. This example also shows that the inequality cu(L)ca(L)c_{u}(L) \leq c_{a}(L) could be strict. Moreover, by a result of M.J. Dias Carneiro [4] these Anosov energy levels do not have minimizing measures. Finally, we describe a large class of Lagrangians for which cu(L) is strictly bigger than the maximum of the energy restricted to the zero section of TM.

Cite this article

G. Paternain, Miguel Paternain, Critical values of autonomous Lagrangian systems. Comment. Math. Helv. 72 (1997), no. 3, pp. 481–499

DOI 10.1007/S000140050029