One-dimensional variational problems

By J. Moser, Oliver Knill. 2003

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Lectures on the free period Lagrangian action functional (2013, background): “ean Science Foundation. 1 2 ALBERTO ABBONDANDOLO and is a solution of the Euler-Lagrange equation associated to L, which is unique whenever the two points are suﬃciently close to each other (see e.g. [BGH98] or [Maz11]). Typical examples of Tonelli Lagrangians are electromagnetic Lagrangians, that i”
Variational principle for contact Hamiltonian systems and its applications (2017, background): “By Tonelli Theorem (see for instance [3]) the above infimum can be achieved.”
Optimal transportation on non-compact manifolds (2007, background): “Once one knows that γ is Lipschitz, when L is C(1) it is possible to differentiate the action, see [10], [14], or [21], and, using an integration by parts, one can show that γ satisfies the following integrated form (IEL’) of the Euler-Lagrange equation for almost every t ∈ [t0, t1], for some fixed ”
Weak kam theorem on non compact manifolds (2007, background): “Since the curves in P ′(x, c, t) are all contained in the closed Riemannian ball centered in x and of radius K(c, t), which is compact by the completeness of the metric, Tonelli’s theory, see [3], [11] or [14], and the continuity of u then shows that the infimum in the definition of T− t u(x) is att”