Approximate Noether Symmetries of Perturbed Lagrangians and Approximate Conservation Laws

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented.

The field-theoretical methods in Lovelock gravity

The field-theoretical methods are used to construct conserved currents and related superpotentials for perturbations on arbitrary backgrounds in the Lovelock gravity. The perturbations are considered as a dynamic field configuration propagating in a given spacetime. The field-theoretical formalism is exact (without approximations) and equivalent to the original metric theory. As Lagrangian based formalism, it allows us to apply the Noether theorem. As a result, we construct conserved currents and superpotentials, where we use arbitrary displacement vectors, not only the Killing ones or other special vectors. The developed formalism is checked in calculating mass of the Schwarzschild-anti-de Sitter (AdS) black hole. The new formalism is adopted to the case of a so-called pure Lovelock gravity, where in the Lagrangian only a one polynomial in Riemannian tensor presents. We construct conserved charges and currents for static and dynamic black holes of the Vaidya type with AdS, dS and flat asymptotics. New properties of the solutions under consideration have been found. The more results are discussed. The first section in your paper

MATERIAL OPTIMIZATION AND DYNAMIC MATERIALS

The paper is about the connection between material optimization in dynamics and a novel concept of dynamic materials (DM) defined as inseparable union of a framework and the fluxes of mass, momentum, and energy existing in time dependent material formations. An example of a spatial-temporal material geometry is discussed as illustration of a DM capable of accumulating wave energy. Finding the optimal material layouts in dynamics demonstrates conceptual difference from a similar procedure in statics. In the first case, the original constituents are distributed in space-time, whereas in the second - in space alone. The habitual understanding of a material as an isolated framework has come from statics, but a transition to dynamics brings in a new component - the fluxes of mass, momentum, and energy. Based on Noether theorem, these fluxes connect the framework with the environment into inseparable entity termed dynamic material (DM). The key role of DM is that they support controls that may purposefully change the material properties in both space and time, which is the main goal of optimization.

Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

We establish a generalization of the Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton–Jacobi–Bellman equation associated with an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton’s optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales.