On the Variational Problems without Having Desired Variational Symmetries

Symmetries of Variational Problems

The applications of symmetry groups to problems arising in the calculus of variations have their origins in the late papers of Lie, e.g., [34], which introduced the subject of “integral invariants”. Lie showed how the symmetry group of a variational problem can be readily computed based on an adaptation of the infinitesimal method used to compute symmetry groups of differential equations. Moreo...

Gauge Symmetries in Fractional Variational Problems

We prove a second Noether theorem for Lagrangian densities with fractional derivatives defined in the Riesz–Caputo sense. An application to the fractional electromagnetic field is given. AMS Subject Classifications: 49K05, 26A33.

Hartley Series Direct Method for Variational Problems

The computational method based on using the operational matrix of anorthogonal function for solving variational problems is computeroriented. In this approach, a truncated Hartley series together withthe operational matrix of integration and integration of the crossproduct of two cas vectors are used for finding the solution ofvariational problems. Two illustrative...

Computing ODE Symmetries as Abnormal Variational Symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [Comput. Methods Appl. Math. 5 (2005), no. 4, pp. 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of i...

Variational Formulation of Problems and Variational Methods