Related papers: Invariantised Euler-Lagrange equations and conserv…
The explicit formulation of the general inverse problem on conservation laws is presented for the first time. In this problem one aims to derive the general form of systems of differential equations that admit a prescribed set of…
A simple procedure to obtain complete, closed expressions for Lie algebra invariants is presented. The invariants are ultimately polynomials in the group parameters. The construction of finite group elements require the use of projectors,…
The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multisymplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is…
The problem of minimizing an integral functional of a vector-valued Lagrangian on a set of admissible arcs with given endpoints is considered. The problem is tackled by embedding it into a set-optimization problem such that the image space…
The one-dimensional modified shallow water equations in Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian coordinates, in mass Lagrangian variables, and Eulerian…
We obtain Euler-Lagrange and transversality optimality conditions for higher-order infinite horizon variational problems on a time scale. The new necessary optimality conditions improve the classical results both in the continuous and…
It is widely accepted that conservation laws, especially energy-momentum conservation, have fundamental importance for both classical and quantum systems in physics. A widely used method to derive the conservation laws is based on Noether's…
The problem of the classical non-relativistic electromagnetically kicked oscillator can be cast into the form of an iterative map on phase space. The original work of Zaslovskii {\it et al} showed that the resulting evolution contains a…
We consider an inverse extremal problem for variational functionals on arbitrary time scales. Using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variational functional that attains a…
The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section,…
In this paper we give a general geometrical framework for working with problems that can be described as a structure-preserving submersion defined on a suitable space with a geometrical structure. We give many examples of how to formulate…
Main results and techniques of the fractional calculus of variations are surveyed. We consider variational problems containing Caputo derivatives and study them using both indirect and direct methods. In particular, we provide necessary…
We summarize some our recent results on encoding exact solutions of field equations in Einstein and modified gravity theories into solitonic hierarchies derived for nonholonomic curve flows with associated bi-Hamilton structure. We argue…
The present paper is devoted to the group classification of magnetogasdynamics equations in which dependent variables in Euler coordinates depend on time and two spatial coordinates. It is assumed that the continuum is inviscid and…
We study the relativistic Euler equations on the Minkowski spacetime background. We make assumptions on the equation of state and the initial data that are relativistic analogs of the well-known physical vacuum boundary condition, which has…
Second gradient theories have to be used to capture how local micro heterogeneities macroscopically affect the behavior of a continuum. In this paper a configurational space for a solid matrix filled by an unknown amount of fluid is…
We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved,…
We deal with homogeneous Dirichlet and Neumann boundary-value problems for anisotropic elliptic operators of p-Laplace type. They emerge as Euler-Lagrange equations of integral functionals of the Calculus of Variations built upon possibly…
We prove a version of the variational Euler-Lagrange equations valid for functionals defined on Fr\'echet manifolds, such as the spaces of sections of differentiable vector bundles appearing in various physical theories.
Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler-Lagrange equations. Using group classification and Noether's theorem,…