Relativistic Maxwell Theory and Einstein Field Equation derived from Variational Principle

In this article, I'll be reviewing relativistic mechanisms using the calculus of variation in the classical limit. The variational principle is considered to be one of the most important mechanisms to build a theory. Newton's second law of motion is a consequence of Euler-Lagrange equations which gives the least (or stationary) trajectory of a particle between any two arbitrary points. I'll the use action principle by deriving the relativistic Maxwell's field equation, geodesic equation, and Einstein's field equation.

Application of Calculus of Variation in the Optimization of Functional Parameters of Compacted Modified Soils: A Simplified Computational Review

Fixed endpoint problems (FEPPs) in constrained systems like the effect of curing time or the effect of certain additives in soil stabilization operations have been reviewed illustratively for sustainability purposes in geotechnics. The calculus of variation (CoV) technique of Hamilton’s problem was demonstrated using a typical case in geotechnics; the effect of curing time on the unconfined compressive strength of expansive soils is utilized as foundation materials. The era of smart technologies is evolving, and to key into this fast-moving area to help the field of geotechnics, it is required that these new areas are deployed to study their usefulness. The use of CoV in modeling or simulating geotechnical properties of soil behavior is not prominent and has been played down due to the uncertainties surrounding it. However, this work has identified that if any geotechnical system can be demonstrated in graphs, then the use of CoV becomes easy with the mathematical concept that curves are elements of straight paths. The results of this work show that CoV is a powerful tool to achieving sustainable optimization of quality properties of stabilized for sustainable and optimal materials handling, design, and construction.

Hamilton-Jacobi Equation of Time Dependent Hamiltonians

In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of time- dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism.

Solutions of the (free boundary) Reifenberg Plateau problem

We solve two variants of the Reifenberg problem for all coefficient groups. We carry out the direct method of the calculus of variation and search a solution as a weak limit of a minimizing sequence. This strategy has been introduced by De Lellis, De Philippis, De Rosa, Ghiraldin and Maggi and allowed them to solve the Reifenberg problem. We use an analogous strategy proved in [C. Labourie, Weak limits of quasiminimizing sequences, preprint 2020, https://arxiv.org/abs/2002.08876] which allows to take into account the free boundary. Moreover, we show that the Reifenberg class is closed under weak convergence without restriction on the coefficient group.

Locomotion generation for a mobile manipulator by global minimization of the weighted generalized momentum

This article presents a method for generating the locomotion of a mobile manipulator that globally minimizes the weighted generalized momentum. The method utilizes the calculus of variation setting to address the problem for which the optimal trajectory may be computed by solving the initial value problem of the system of ordinary differential equations rather than the two-point boundary value problem. Online optimal trajectory may then be input to a suitable tracking controller for controlling the robot in real time. Effectively, the robot closed-loop dynamics is shaped to the optimal system such that the locomotion minimizes the difference of the weighted generalized momentum and the assigned potential energy under the constraints imposed on by the tracking task, joint angle, and actuator torque limits. Desired locomotion behaviors may be achieved by properly adjusting the weighting, spring, and damping matrices. Exploiting the induced dynamical force from the cooperative motion of the constituent linkages through the momentum minimization basis, the robot is able to outperform conventional locomotion pattern actuated by the platform solely.

On control reconstructions to management problems

We present and discuss a new approach to solutions of control reconstruction problems in real time. The suggested solution based on necessary optimality conditions for auxiliary calculus of variation problems with concave-convex discrepancy functional.

Bernoulli Collocation Polynomials Algorithm for Calculus of Variational Problems

<p>This paper presents an approximate method that depends on the Bernoulli Polynomials as basic functions. The method is concerned with collocation technique for solving problems in calculus of variation. Some interesting properties of Bernoulli polynomials are used to reduce the original problem to mathematical problem. Some illustrative examples are described to show the applicability of the proposed method.</p>