Showing papers on "Noether's theorem published in 1977"

Ambiguities in the Lagrangian and Hamiltonian formalism: Transformation properties

Abstract: It is demonstrated that the classical isotropic oscillator can be described by simple Lagrangians and Hamiltonians which do not have the symmetry properties usually associated with these functions For instance, Noether's theorem need not connect angular momentum with rotations By starting from this example, the relation between symmetries and constants of the motion is analysed in terms of vector fields on symplectic manifolds in the modern view of classical dynamics This relation is shown to be far more ambiguous than is usually thought

Constants of the Motion in Lagrangian Mechanics

Abstract: Two points of view of the relationship between symmetries of a dynamical system and constants of the motion, in the Lagrangian framework, are compared. The first point of view is that associated with Noether's theorem, the second, with the Cartan form approach to Lagrangian mechanics. It is argued that the second is more satisfactory.

Noether’s theorem in classical mechanics

Abstract: A simple statement of Noether’s theorem as it applies to classical mechanics is given and proved. It is then used to generate constants of the motion associated with Lagrangians possessing certain transformation properties.

Time‐dependent dynamical symmetries and constants of motion. III. Time‐dependent harmonic oscillator

Abstract: This paper is a continuation of previous Papers I and II [J. Math. Phys. 17, 1345 (1976); 18, 424 (1977)]. In the present paper we apply the theory (based upon Lagrangian dynamics) developed in I and II to obtain the dynamical symmetries and concomitant constants of motion admitted by the time‐dependent n‐dimensional oscillator (a) Ei≡χi +2ω (t) xi=0. The dynamical symmetries are based upon infinitesimal transformations of the form (b) χi=xi+δxi, δxi≡ξi(x,t) δa; t=t+δt, δt≡ξ0(x,t) δa which satisfy the condition (c) δEi=0, whenever Ei=0. It is shown that such symmetries of the oscillator (a) will be time‐dependent projective collineations. For such symmetries which satisfy the R1 restriction (defined in I) it is shown there exist concomitant constants of motion C1 of the oscillator, which for n=1 are time‐dependent cubic polynomials in the χ variable, and for n⩾2 are time‐dependent quadratic polynomials in the χi variables. It is shown that those symmetries which satisfy the R2 restriction (Noether symmetry condition discussed in I) are time‐dependent homothetic mappings consisting of time‐dependent scale changes, time‐dependent translations, and rotations. The concomitant Noether constants of motion C2 are time‐dependent quadratic polynomials in the χi variables for all n. The Noether constant of motion C2 [referred to as C2(B)] for which the associated underlying symmetry mapping is the time‐dependent scale change is shown to include as a special case when n=1 a class of invariants formulated by Lewis [Phys. Rev. Lett. 18, 510 1967)] (by means of a phase space analysis which applies Kruskal’s theory in closed form). For the case of general n it is shown that the time‐dependent symmetric tensor constant of motion Iij constructed by Gunther and Leach [J. Math. Phys. 18, 572 (1977)] is included as a special case of a time‐dependent symmetric tensor constant of motion Kij, where Kij is obtained by use of a time‐dependent related integral theorem by means of the symmetry deformation of the constant of motion C2(B) with respect to the affine collineations; such collineations are a subset of the projective collineation symmetries mentioned above. The symmetries and their concomitant constants of motion of the oscillator (a) with ω (t) of the form ω (t) =a+bect are obtained.

On stress-energy tensors

Abstract: We show how to introduce the “Noether Operator” of a (possibly constrained) variational principle even when the Lagrangian contains spinor fields (and their derivatives to any finite order). After relating that operator to the so-called “canonical” and “symmetric” stress-energy tensors, we construct explicitly the divergence by which these differ. A brief appendix illustrates the method of dealing with spinors by calculating Tμv for the Dirac equation.

Nonelectromagnetic interactions from galilei invariance

Abstract: A realistic dynamics of a massive pointlike Newtonian particle is constructed, by imposing the invariance under the localized pure Galilean boosts. A direct use is made of the second Noether theorem, whose constraint identities give a clue for the construction of the gauge Lagrangian. The obtained interaction shares gauge invariance properties different from the electromagnetic ones; we show it is possible to give a purely kinematical—broadly speaking—interpretation of the results. The corresponding quantum mechanics can be realized with strictly invariant wave functions and operators belonging to true-Galilei-algebra faithful realizations.

Conformal identities for invariant second-order variational problems depending on a covariant vector field

Abstract: Necessary and sufficient conditions for the conformal invariance of a multiple integral variational problem whose Lagrangian depends upon second-order derivatives of a covariant vector field are obtained. These conditions take the form of differential identities involving the Lagrangian, its derivatives, and the infinitesimal generators of the special conformal group; they differ from the classical Noether identities in that they involve only second-order derivatives of the field, not fourth-order derivatives. The conditions are not conservation laws, but rather identities which provide a practical test for invariance which, if established, can lead to conservation laws via the Noether theorem. Finally, an application to 'generalised electrodynamics' is given.