Dynamical symmetries and constants of the motion for classical particle systems

Abstract

By formulating the conditions for dynamical symmetry mappings directly at the level of the dynamical equations (which are taken in the form of Newton's equations, Lagrange's equations, Hamilton's equations, or Hamilton-Jacobi equation), we derive new expressions for dynamical symmetries and associated constants of the motion for classical particle dynamical systems. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (a) x̄i =xi+δxi [δxi≡ξi(x)δa] with associated changes in the independent variable t (path parameter) defined by (b) δt≡{∫2φ[x(t)]dt+c} δa. A generalized form of the related integral theorem (a method for obtaining constants of the motion based upon deformations of a known constant of the motion under dynamical symmetry mappings) is obtained. We take the ``Newtonian form'' of the dynamical equations to have a coordinate-covariant structure with forces defined by a general polynomial in the velocities and obtain dynamical symmetry conditions for all such systems. For the special case of conservative systems the related integral theorem is applied. Based upon Lagrange's equations with L=L(xi,ẋi) we find the conditions for dynamical symmetry mappings may be expressed in the form (c) (∂/∂xj)[δL+L(d/dt)(δt)]−(d/dt)(∂/∂ẋj)[δL+L(d/dt)(δt)]=−2φ,j[(∂L/∂ẋi)ẋi−L]δa. From this form we obtain a new formula for concomitant constants of the motion: (d) [∂(δL)/∂ẋj] ẋj −δL = k. By use of the related integral theorem such constants of the motion can be expressed as deformations of the energy integral under the dynamical symmetry mappings defined by (c). A short derivation of the Noether identity is given which is independent of the integration processes of Hamilton's variational principle. For mappings of the type (a), (b) ``Noether type'' symmetries and associated constants of the motion are formulated. For a conservative dynamical system with L≡(1/2)gijẋiẋj − V(x) we find such Noether symmetries are basically conformal motions, while those derived from (c) are basically projective collineations. For such systems the constants of the motion (d) are evaluated and shown in general to differ from those obtained from the Noether method. We show for conservative dynamical systems that the formulation of dynamical symmetry mappings directly at the level of the Hamilton-Jacobi equation leads to the Noether symmetry conditions. Dynamical symmetry conditions are formulated for Hamilton's equation in phase space and shown to be more general than canonical transformations. The formulation of the related integral theorem in phase space is found to be a generalization of Poisson's theorem. For systems with H(xA), A = 1,…,2n, it is an immediate observation that δH induced by a symmetry mapping is a constant of the motion. Application to the isotropic harmonic oscillator shows both symmetric tensor and angular momenta constants of the motion are obtained in this manner. An additional constant of the motion ∂A ξA−2φ(xA) is shown in general to be a concomitant of a phase space symmetry transformation.