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On minimizing measures of the action of autonomous Lagrangians. (English) Zbl 0845.58023
Using the theory of J. N. Mather [Math. Z. 207, No. 2, 169–207 (1991; Zbl 0716.58020)] the author describes some dynamical properties of trajectories of the Euler-Lagrange flow associated to the Lagrangian $$L$$ which are contained in the support of a probability $$\mu$$ which have a given rotation vector and minimizes the action function $$A_L(\mu)=\int L\,d\mu$$. It is shown that the minimal action function $$\beta(h)=\min\{\int L\,d\mu:\rho(\mu)=h\}$$, has radial derivative, (where $$\rho(\mu)\in H_1(M,\mathbb R)$$ is the homology vector of the invariant probability $$\mu$$) and that there exists a one-to-one correspondence between minimizing measures for Lagrangians associated to mechanical systems and minimizing measures for the geodesic problem on a fixed energy level.

##### MSC:
 58E30 Variational principles in infinite-dimensional spaces 37J50 Action-minimizing orbits and measures (MSC2010)
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