We extend the DuBois--Reymond necessary optimality condition
and Noether's first theorem to variational problems of Herglotz
type with time delay. Our results provide, as corollaries,
the DuBois--Reymond necessary optimality condition and the first
Noether theorem for variational problems with time delay recently
proved in [Numer. Algebra Control Optim. 2 (2012), no. 3, 619--630].
Our main result is also a generalization of the first Noether-type theorem
for the generalized variational principle of Herglotz proved in
[Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 261--273].

Variational problems of Herglotz typewith time delay: DuBois--Reymond condition and Noether's first theorem

We obtain a generalized Euler–Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.

/pdf/higher-order-variational-problems-of-herglotz-type-4dxrickfmi.pdf

Higher-order variational problems of Herglotz type

We obtain a generalized Euler-Lagrange differential equation and transversality optimality conditions for Herglotz-type higher-order variational problems. Illustrative examples of the new results are given.

The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied. This type of problem generalizes several problems recently studied in the literature. Moreover, it allows us to unify conservative and non-conservative dynamical processes in the same model. The dependence of the Lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard Herglotz’s variational problem into a problem of a different nature.

Fractional variational principle of Herglotz for a new class of problems with dependence on the boundaries and a real parameter

The main goal of this paper is to extend the generalized variational problem of Herglotz type to the more general context of the Euclidean sphere Sn. Motivated by classical results on Euclidean spaces, we derive the generalized Euler–Lagrange equation for the corresponding variational problem defined on the Riemannian manifold Sn. Moreover, the problem is formulated from an optimal control point of view and it is proved that the Euler–Lagrange equation can be obtained from the Hamiltonian equations. It is also highlighted the geodesic problem on spheres as a particular case of the generalized Herglotz problem.

/pdf/the-herglotz-variational-problem-on-spheres-and-its-optimal-4ha81denp2.pdf

The Herglotz variational problem on spheres and its optimal control approach

The Herglotz algorithm for constructing autonomous canonical transformations that was presented by Guenther, Gottsch, and Kramer in [SIAM Rev., 38 (1996), pp. 287--293] is extended to include nonautonomous canonical transformations and a remainder function ${\cal R}^*$ that is related to a generating function by ${\cal R}^* = \partial F/\partial t$.

Extension of the Herglotz Algorithm to Nonautonomous Canonical Transformations

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Extension of the Herglotz Algorithm to Nonautonomous Canonical Transformations