Conserved quantities in the variational equations

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).

Il Nuovo Cimento A, 1970

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

Physical Review D, 2000

In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define "conserved quantities" in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired "conserved quantities" are not, in general, conserved!) In this paper we give a prescription for defining "conserved quantities" by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, 1 although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations.

2015

This paper expounds the relations between continuous symmetries and con-served quantities, i.e. Noether’s “first theorem”, in both the Lagrangian and Hamiltonian frameworks for classical mechanics. This illustrates one of mechan-ics ’ grand themes: exploiting a symmetry so as to reduce the number of variables needed to treat a problem. I emphasise that, for both frameworks, the theorem is underpinned by the idea of cyclic coordinates; and that the Hamiltonian theorem is more powerful. The Lagrangian theorem’s main “ingredient”, apart from cyclic coordinates, is the rectification of vector fields afforded by the local existence and uniqueness of solutions to ordinary differential equations. For the Hamiltonian theorem, the main extra ingredients are the asymmetry of the Poisson bracket, and the fact that a vector field generates canonical transformations iff it is Hamiltonian. 1email:

2003

Using a theorem of partial differential equations, we present a general way of deriving the conserved quantities associated with a given classical point mechanical system, denoted by its Hamiltonian. Some simple examples are given to demonstrate the validity of the formulation.

Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics, 2015

There is general consensus among physicists in considering symmetries as a source of conserved quantities, a conclusion allegedly supported by Emmy Noether's theorems. Recently it has been pointed out that no arrow of explanation can be extracted from Noether's work, and there are also criticisms against the priority of particular symmetries over specific conserved quantities under Noether's ideas, but there are no general arguments against the aforementioned consensus, nor proposals promoting an explanation that leads from conserved quantities to symmetries. In this paper a general argument is built which favours conserved quantities over symmetries inasmuch as the presence of the former seems to allow (i.e. it seems to be a sufficient condition leading to) symmetrical descriptions.

International Journal of Geometric Methods in Modern Physics, 2005

Augmented variational principles are introduced in order to provide a definition of relative conservation laws. As it is physically reasonable, relative conservation laws define in turn relative conserved quantities that measure, for example, how much energy is needed in a field theory to go from one configuration (called the reference or vacuum) to another configuration (the physical state of the system). The general prescription we describe solves in a covariant way the well-known observer dependence of conserved quantities. The solution found is deeply related to the divergence ambiguity of the Lagrangian and to various formalisms that have recently appeared in literature to deal with the variation of conserved quantities (of which this is a formal integration). A number of examples relevant to fundamental physics are considered in detail, starting from classical mechanics.

2019

In General Relativity (GR) finding out geodesics of a given spacetime manifold is an important task because it determines what are the classical processes which are dynamically forbidden. Conserved quantities play an important role in solving geodesic equations of a general spacetime manifold. Furthermore, knowing all possible conserved quantities of a system tells about the hidden symmetries of a system which is not apparent, since conserved quantities are deeply connected with symmetry properties of the system, which are very important in their own right. This connection is more transparent due to Noether's theorem with the development of the Lagrangian and Hamiltonian formulations of Newtonian mechanics. Conserved quantities are also useful to capture certain features of spacetime manifold for an asymptotic observer. In this report, we have found out the possible methods of finding conserved quantities and their algebra in the dynamics of an object in curved spacetime both in presence and absence of spin of that object which will help in analyzing dynamics of objects in curved spacetime in future run of advanced LIGO with the help of gravitational memory effect and gravitational lensing effect using principles of electromagnetism and GR.

Certain features of the works of Dyson (Am. are integrated with our definition of the Poisson bracket on a tangent bundle (velocity phase space ) to address the Noether theorem inversion and inverse problem in the Newtonian mechanics from a novel viewpoint without assuming apriori a Lagrangian or Hamiltonian . From this standpoint we study the existence, uniqueness and form of the Lagrangian, and, how to pass from conservation laws to Noether symmetries.