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By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system…
Making use of the Lagrange anchor construction introduced earlier to quantize non-Lagrangian field theories, we extend the Noether theorem beyond the class of variational dynamics.
We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same…
The concept of nonlinear self-adjointness is employed to construct the conservation laws for fractional evolution equations using its Lie point symmetries. The approach is demonstrated on subdiffusion and diffusion-wave equations with the…
Two applications of the Noether method for fluids and plasmas are presented based on the Euler-Lagrange and Euler-Poincare variational principles, which depend on whether the dynamical fields are to be varied independently or not,…
Any conformally invariant energy associated with a curve possesses tension-free equilibrium states which are self-similar. When this energy is the three dimensional conformal arc-length, these states are the natural spatial generalizations…
We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the…
A new regularisation of the shallow water (and isentropic Euler) equations is proposed. The regularised equations are non-dissipative, non-dispersive and possess a variational structure. Thus, the mass, the momentum and the energy are…
We study general metric-affine theories of gravity in which the metric and connection are the two independent fundamental variables. In this framework, we use Lagrange-Noether methods to derive the identities and the conservation laws that…
The Noether theorem connecting symmetries and conservation laws can be applied directly in a Hamiltonian framework without using any intermediate Lagrangian formulation. This requires a careful discussion about the invariance of the…
The present paper is focused on the analysis of the one-dimensional relativistic gas dynamics equations. The studied equations are considered in Lagrangian description, making it possible to find a Lagrangian such that the relativistic gas…
We combine the construction of the canonical conservation law and the nonlocal cosymmetry to derive a collection of nonlocal conservation laws for the two-dimensional Euler equation in vorticity form. For computational convenience and…
This paper mainly contributes to the extension of Noether's theorem to differential-difference equations. For that purpose, we first investigate the prolongation formula for continuous symmetries, which makes a characteristic representation…
In the Lagrangian framework for symmetries and conservation laws of field theories, we investigate globality properties of conserved currents associated with non-global Lagrangians admitting global Euler--Lagrange morphisms. Our approach is…
The aim of this note is to discuss the relation between one-parameter continuous symmetries of the dynamics, defined on physical grounds, and conservation laws. In the Hamiltonian formulation, such symmetries of the dynamics in general…
For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is…
The recently discovered conserved quantity associated with Kepler rescaling is generalised by an extension of Noether's theorem which involves the classical action integral as an additional term. For a free particle the familiar…
We consider a new variant of cosmological perturbation theory that has been designed specifically to include non-linear density contrasts on scales 100 Mpc, while still allowing for linear fluctuations on larger scales. This theory is used…
A fifth-order KdV equation with time dependent coefficients and linear damping has been studied. Symmetry groups have several different applications in the context of nonlinear differential equations. For instance, they can be used to…
This paper presents recent work on connections between symmetries and conservation laws. After reviewing Noether's theorem and its limitations, we present the Direct Construction Method to show how to find directly the conservation laws for…