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Lagrangian flows: the dynamics of globally minimizing orbits. II. (English) Zbl 0892.58065
Let the critical level \(c(L)\) of a convex superlinear Lagrangian \(L\) be the infimum of reals \(k\) such that the Lagrangian \(L+k\) has minimizers with fixed endpoints and free time interval.
The authors provide the proofs of theorems characterizing \(c(L)\) in terms of minimizing measures of \(L\). There are proved results for cohomology properties for minimizers of \(L+c(L)\). The following Tonelli’s theorem is proved: There exist minimizers of the \(L+k\)-action joining any two points in the projection of \(E=k\) among curves with energy \(k\). [For Part I, see R. Mañé, ibid., 141-153 (1997; Zbl 0892.58064) see the review above].
Reviewer: S.Nenov (Sofia)

MSC:
37C10 Dynamics induced by flows and semiflows
37A99 Ergodic theory
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References:
[1] R. Abraham & J. E. Marsden. Foundations of Mechanics. Benjamin: London, 1978. · Zbl 0393.70001
[2] M. J. Dias Carneiro.On Minimizing Measures of the Actions of Autonomous Lagrangians. Nonlinearity,8 (1995) no. 6, 1077-1085. · Zbl 0845.58023
[3] G. Contreras, R. Iturriaga.Convex Hamiltonians without conjugate points. Preprint. Available via internet at http://www.ma.utexas.edu. · Zbl 1044.37046
[4] R. Mañé. Global Variational Methods in Conservative Dynamics. 18{\({}^o\)} Coloquio Bras. de Mat. IMPA. Rio de Janeiro, 1991.
[5] R. Mañé.On the minimizing measures of Lagrangian dynamical systems. Nonlinearity,5, (1992), no.3, 623-638. · Zbl 0799.58030
[6] R. Mañé.Generic properties and problems of minimizing measures of lagrangian systems. Nonlinearity,9, (1996), no.2, 273-310. · Zbl 0886.58037
[7] R. Mañé.Lagrangian Flows: The Dynamics of Globally Minimizing Orbits. In International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé), F. Ledrappier, J. Lewowicz, S. Newhouse eds., Pitman Research Notes in Math.362 (1996) 120-131. Reprinted in Bol. Soc. Bras. Mat. Vol28, N. 2, 141-153. · Zbl 0870.58026
[8] J. Mather.Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z.207, (1991), no. 2, 169-207. · Zbl 0696.58027
[9] M. Morse. Calculus of variations in the large. Amer. Math. Soc. Colloquium Publications vol. XVIII, 1934. · Zbl 0011.02802
[10] S. Schwartzman.Asymptotic cycles. Ann. of Math. (2)66 (1957) 270, 284. · Zbl 0207.22603
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