We prove Euler-Lagrange fractional equations and sufficient optimality conditions for problems of the calculus of variations with functionals containing both fractional derivatives and fractional integrals in the sense of Riemann-Liouville.

This paper presents the Euler–Lagrange equations for fractional variational problems with multiple integrals. The fractional Noether-type theorem for conservative and nonconservative generalized physical systems is proved. Our approach uses well-known notion of the RiemannLiouville fractional derivative.

2 The Calculus of Variations 2 2.1 Functionals of f and f ′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.2 Functionals of f , f ′, and f ′′ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Cubic Splines and Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Euler-Lagrange Equations for Multi...

We present an approach to the canonical quantization of systems with equations of motion that are historically called non-Lagrangian equations. Our viewpoint of this problem is the following: despite the fact that a set of differential equations cannot be directly identified with a set of Euler-Lagrange equations, one can reformulate such a set in an equivalent first-order form which can always...

In this paper, we review two related aspects of field theory: the modeling fields by means exterior algebra and calculus, derivation dynamics, i.e., Euler-Lagrange equations, stationary action principle. contrast to usual tensorial these equations for theories, that gives separate components, coordinate-free forms are derived. These alternative reminiscent formulae vector expressed in terms der...

In this paper we develop a fractional Hamiltonian formulation for dynamic systems defined in terms of fractional Caputo derivatives. Expressions for fractional canonical momenta and fractional canoni-cal Hamiltonian are given, and a set of fractional Hamiltonian equations are obtained. Using an example, it is shown that the canonical fractional Hamiltonian and the fractional Euler-Lagrange form...

This paper reviews the moving frame approach to the construction of the invariant variational bicomplex. Applications include explicit formulae for the Euler-Lagrange equations of an invariant variational problem, and for the equations governing the evolution of differential invariants under invariant submanifold flows.

Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constrain...

The dynamics of randomly forced Burgers and Euler-Lagrange equations in S 1 × R d−1 in the case when there is only one one-sided minimizer in a compact subset of S 1 × R d−1 is studied. The existence of random invariant periodic minimizer orbits and periodicity of the stationary solution of the stochastic Burgers equations are obtained.

In a recent paper [7] we interpreted configurational forces as necessary and sufficient dissipative mechanisms such that the corresponding Euler-Lagrange equations are satisfied. We now extend this argument for a dynamic elastic medium, and show that the energy flux obtained from the dynamic J integral ensures that the equations of motion hold throughout the body.