Fractional (or non-integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject, the main result being the fractional necessary optimality condition of Euler-Lagrange obtained in 2002. Here we use the notion of Euler-Lagrange fractional extremal to prove a Noether-type theorem. For that we propose a generalization of the classical concept of conservation law, introducing an appropriate fractional operator.

Using the recent formulation of Noether's theorem for the problems of the calculus of variations with fractional derivatives, the Lagrange multiplier technique, and the fractional Euler-Lagrange equations, we prove a Noetherlike theorem to the more general context of the fractional optimal control. As a corollary, it follows that in the fractional case the autonomous Hamiltonian does not define anymore a conservation law. Instead, it is proved that the fractional conservation law adds to the Hamiltonian a new term which depends on the fractional-order of differentiation, the generalized momentum, and the fractional derivative of the state variable.

We prove a Noether's theorem for fractional variational problems with Riesz-Caputo derivatives. Both Lagrangian and Hamiltonian formulations are obtained. Illustrative examples in the fractional context of the calculus of variations and optimal control are given.

We extend the DuBois-Reymond necessary optimality condition and Noether's symmetry theorem to the scale relativity theory setting. Both Lagrangian and Hamiltonian versions of Noether's theorem are proved, covering problems of the calculus of variations with functionals defined on sets of non-differentiable functions, as well as more general non-differentiable problems of optimal control. As an application we obtain constants of motion for some linear and nonlinear variants of the Schrödinger equation.In this section we briefly review the quantum calculus of [3], which extends the classical differential calculus to non-differentiable functions.We denote by C 0 the set of real-valued continuous functions defined on R.Definition 1. Let f ∈ C 0 . For all ǫ > 0, the ǫ left-and right-quantum deriva-

In the present work, we propose an Action Principle for Action-dependent Lagrangians by generalizing the Herglotz variational problem for several independent variables. This Action Principle enables us to formulate Lagrangian densities for non-conservative fields. In special, from a Lagrangian depending linearly on the Action, we obtain a generalized Einstein's field equations for a non-conservative gravity and analyze some consequences of their solutions to cosmology and gravitational waves. We show that the non-conservative part of the field equations depends on a constant cosmological four-vector. Depending on this four-vector, the theory displays damped/amplified gravitational waves and an accelerating Universe without dark energy.The Action Principle was introduced in its mature formulation by Euler, Hamilton and Lagrange and, since then, it has become a fundamental principle for the construction of all physical theories. In order to obtain the dynamical equations of any theory, the Lagrangian defining the Action is constructed from the scalars of the theory. In this case, the action itself is a scalar. Consequently, we might ask: what would happen if the Lagrangian itself is a function of the Action? The answer to this question can be given by the Action Principle proposed by Herglotz [1][2][3]. The Herglotz variational calculus consists in the problem of determining the path x(t) that extremize (minimize or maximize) S(b), where S(t) is a solution oḟIt is easy to note that (1) represents a family of differential equations since for each function x(t) a different differential equation arises. Therefore, S(t) is a functional. The problem reduces to the classical fundamental problem of the calculus of variations if the Lagrangian function L does not depend on S(t). In this case we havė S(t) = L(t, x(t),ẋ(t)), and by integrating we obtain the classical variational problemwhereIt is important to notice from (2) that for a given fixed function x(t) the functional S reduces to a function of the domain boundary a, b. Herglotz proved [1, 2] that a necessary condition for a path x(t) to be an extremizer of the variational problem (1) is given by the generalized Euler-Lagrange equation:In the simplest case where the dependence of the Lagrangian function on the Action is linear, the Lagrangian describes a dissipative system and, from (4), the resulting equation of motion includes the well known dissipative term proportional toẋ. It should also be noticed that in the case of the classical problem of the calculus of variation (2) one has ∂L ∂S = 0, and the differential equation (4) reduces to the classical Euler-Lagrange equation.