On Symmetry and Conserved Quantities in Classical Mechanics (original) (raw)
Related papers
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:
On the Derivation of Conserved Quantities in Classical Mechanics
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.
Symmetries in Non-Linear Mechanics
In this paper we exploit the use of symmetries of a physical system so as to characterize the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantisation in non-linear cases, where the success of Canonical Quantisation is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order. The use of the Poincaré-Cartan form permits finding both the symplectic structure on the solution manifold, through the Hamilton-Jacobi transformation, and the required symmetries, realized as Hamiltonian vector fields, associated with functions on the solution manifold (thus constituting an inverse of the Noether Theorem), lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. In this framework, solutions and symmetries are somehow identified and this correspondence is also kept at a perturbative level. We present simple non-trivial examples of this interplay between symmetries and solutions pointing out the usefulness of this mechanism in approaching the corresponding quantisation.
ON THE SYMMETRIES OF HAMILTONIAN SYSTEMS
International Journal of Modern Physics A, 1995
In this paper we show how the well-known local symmetries of Lagrangean systems, and in particular the diffeomorphism invariance, emerge in the Hamiltonian formulation. We show that only the constraints which are linear in the momenta generate transformations which correspond to symmetries of the corresponding Lagrangean system. The nonlinear constraints (which we have, for instance, in gravity, supergravity and string theory) rather generate the dynamics of the corresponding Lagrangean system. Only in a very special combination with "trivial" transformations proportional to the equations of motion do they lead to symmetry transformations. We reveal the importance of these special "trivial" transformations for the interconnection theorems which relate the symmetries of a system with its dynamics. We prove these theorems for general Hamiltonian systems. We apply the developed formalism to concrete physically relevant systems and in particular those which are diffeomorphism invariant. The connection between the parameters of the symmetry transformations in the Hamiltonian-and Lagrangean formalisms is found. The possible applications of our results are discussed.
On symmetries in Galilei classical mechanics
2000
In the framework of Galilei classical mechanics (i.e., general relativistic classical mechanics on a spacetime with absolute time) developed by Jadczyk and Modugno, we analyse systematically the relations between symmetries of the geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on spacetime and of its potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism, Lagrangians. Then, we provide a covariant momentum map associated with a group of cosymplectic symmetries by using a covariant lift of functions of phase space. In the case of an action that projects on spacetime we see that the components of this momentum map are quantisable functions in the sense of Jadczyck and Modugno. Finally, we illustrate the results in some examples.
A new approach to the converse of Noether's theorem
Journal of Physics A: Mathematical and General, 1989
The concepts of vector fields and forms along a map are used to establish a condition characterizing symmetries of the Hamiltonian system associated to a regular Lagrangian. This condition does not mention any second order differential equation field but is expressed in terms of the geometry of the second order tangent bundle. This result is also generalized to the case of Lagrangian functions depending on higher order derivatives.
Symmetries in covariant classical mechanics
2000
In the framework of covariant classical mechanics (i.e. general relativistic classical mechanics on a spacetime with absolute time), developed by Jadczyk and Modugno, we analyse systematically the relationship between symmetries of geometric objects. We show that the (holonomic) infinitesimal symmetries of the cosymplectic structure on spacetime and of its horizontal potentials are also symmetries of spacelike metric, gravitational and electromagnetic fields, Euler-Lagrange morphism and Lagrangians. Then, we provide a definition for a covariant momentum map associated with a group of cosymplectic symmetries by means of a covariant lift of functions of phase space. In the case of holonomic symmetries, we see that the any covariant momentum map takes values in the quantizable functions in the sense of Jadczyk and Modugno, i.e. functions quadratic in velocities with leading coefficient proportional to the spacelike metric. Finally, we illustrate the results by some examples.
ON SECOND NOETHER'S THEOREM AND GAUGE SYMMETRIES IN MECHANICS
2000
We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to the process of stabilization of constraints, both in Lagrangian and Hamiltonian formalisms. The connections between both formalisms are established by means of the time-evolution operator.