Symmetries and conservation laws

Journal of Mathematical Physics, 1996

An identity is derived which yields a correspondence between symmetries and conservation laws for self-adjoint differential equations. This identity does not rely on use of a Lagrangian as needed to obtain conservation laws by Noether's theorem. Moreover, unlike Noether's theorem, which can only generate conservation laws from local symmetries, the derived identity generates conservation laws from nonlocal as well as local symmetries. It is explicitly shown how Noether's theorem is extended by the identity. Conservation laws arising from nonlocal symmetries are obtained for a class of scalar wave equations with variable wave speeds. The constants of motion resulting from these nonlocal conservation laws are shown to be linearly independent of all constants of motion resulting from local conservation laws. © 1996 American Institute of Physics. ͓S0022-2488͑96͒02405-2͔ I. INTRODUCTION Conservation laws can be found for self-adjoint systems of differential equations by Noether's theorem. 1-3 If a local symmetry admitted by a given system leaves invariant the variational principle of the system, Noether's theorem yields a corresponding conservation law of local type. Conversely, all conservation laws of local type for a given system arise from the local symmetries admitted by the system. A limitation of Noether's theorem, however, is that it can only directly deal with local symmetries and hence conservation laws of local type. This poses a significant incompleteness in the study of differential equations since conservation laws of nonlocal type are equally as useful as those of local type. In particular, as will be shown in this article, conservation laws of nonlocal type yield additional constants of motion and thus expand the utility of methods of analysis which depend on conservation laws. In this article we introduce an expression that yields conservation laws from nonlocal symmetries as well as local symmetries admitted by an arbitrary self-adjoint system of differential equations. Significantly, in contrast to the formulation of Noether's theorem, the expression is derived from a bilinear identity that makes no use of a Lagrangian. As preliminaries to the derivation and main results, we now give definitions of local and nonlocal symmetries and conservation laws of local and nonlocal type for self-adjoint systems of differential equations. Consider a system of differential equations ͑DEs͒ given by

Theoretical and Mathematical Physics, 1982

Discrete & Continuous Dynamical Systems - S, 2017

Conservation laws are fomulated for systems of differential equations by using symmetries and adjoint symmetries, and an application to systems of evolution equations is made, together with illustrative examples. The formulation does not require the existence of a Lagrangian for a given system, and the presented examples include computations of conserved densities for the heat equation, Burgers' equation and the Korteweg-de Vries equation.

2010

We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories" that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincaré-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether symmetries", energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-called ADM laws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation "à la Palatini" and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).

Journal of Applied Mathematics, 2012

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.

Journal of AppliedMath, 2023

In this study, we investigated a set of equations that exhibit compact solutions and nonlinear dispersion. We used the classical lie symmetry approach to derive ordinary differential equations (ODEs) that are well suited for qualitative study. By examining the dynamic behavior of these ODEs, we gained insights into the intricate nature of the underlying system. We also used a powerful multiplier approach to establish nontrivial conservation laws and exact solutions for these equations. These conservation laws provide essential information regarding the underlying symmetries and invariants of the system, and shed light on its fundamental properties.

Nonlinear Dynamics, 2006

We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.

European Journal of Applied Mathematics, 2017

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, the b-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

Symmetry, 2010

We review the Lagrangian formulation of (generalised) Noether symmetries in the framework of Calculus of Variations in Jet Bundles, with a special attention to so-called "Natural Theories" and "Gauge-Natural Theories" that include all relevant Field Theories and physical applications (from Mechanics to General Relativity, to Gauge Theories, Supersymmetric Theories, Spinors, etc.). It is discussed how the use of Poincaré-Cartan forms and decompositions of natural (or gauge-natural) variational operators give rise to notions such as "generators of Noether symmetries", energy and reduced energy flow, Bianchi identities, weak and strong conservation laws, covariant conservation laws, Hamiltonian-like conservation laws (such as, e.g., so-called ADM laws in General Relativity) with emphasis on the physical interpretation of the quantities calculated in specific cases (energy, angular momentum, entropy, etc.). A few substantially new and very recent applications/examples are presented to better show the power of the methods introduced: one in Classical Mechanics (definition of strong conservation laws in a frame-independent setting and a discussion on the way in which conserved quantities depend on the choice of an observer); one in Classical Field Theories (energy and entropy in General Relativity, in its standard formulation, in its spin-frame formulation, in its first order formulation "à la Palatini" and in its extensions to Non-Linear Gravity Theories); one in Quantum Field Theories (applications to conservation laws in Loop Quantum Gravity via spin connections and Barbero-Immirzi connections).