Gerald H. Katzin

Geodesic first integrals with explicit path‐parameter dependence in Riemannian space–times

In a Riemannian space Vn general formulas are obtained for geodesic first integrals which are mth order polynomials in the tangent vector and which are assumed to depend explicitly on the path parameter s. It is found that such first integrals must also be polynomials in s. Necessary and sufficient conditions are found for the existence of these first integrals. The existence of many well‐known symmetries such as homothetic motions (scale change), affine collineations, conformal motions, projective collineations, conformal collineations, or special curvature collineations are shown to be sufficient for the existence of such first integrals with explicit path‐parameter dependence. To illustrate the theory, geodesic first integrals of this type have been calculated for four Riemannian space–times of general relativity.

Time‐dependent vector constants of motion, symmetries, and orbit equations for the dynamical system r̈=îr{[Ü(t)/U(t)]r −[μ0/U(t)]r−2}

The most general time‐dependent, central force, classical particle dynamical systems (in n‐dimensional Euclidean space, n=2 or 3) of the form (a) r=ir F(r, t), (r2≡r ⋅ r, r=ikxk, k=1,...,n), which admit vector constants of motion of the form (b) I=U(r, t)(L×v)+Z(r, t)(L×r) +W(r, t)r (L≡r×v, v≡r) are obtained. It is found that the only class of such dynamical systems is (c) r=ir(UU−1r−μ0U−1r−2), for which the concomitant vector constant of motion (b) takes the form (d) I=U(L×v)−U(L×r)+μ0r−1r, where in (c) and (d) U=U(t) is arbitrary (≠0). The dynamical system (c) includes both the time‐dependent harmonic oscillator and a time‐dependent Kepler system. Based upon infinitesimal velocity‐independent mappings the complete symmetry group for the dynamical system (c) is obtained. This complete group of [2+n(n−1)/2] parameters contains a complete Noether symmetry subgroup of [1+n(n−1)/2] parameters. In addition to the n(n−1)/2 angular momenta, there is an energy‐like constant of motion also associated with...

A gauge invariant formulation of time‐dependent dynamical symmetry mappings and associated constants of motion for Lagrangian particle mechanics. I

In this paper (part I of two parts), which is restricted to classical particle systems, a study is made of time‐dependent symmetry mappings of Lagrange’s equations (a) Λi(L) =0, and the constants of motion associated with these mappings. All dynamical symmetry mappings we consider are based upon infinitesimal point transformations of the form (b) χi=xi+δxi [δxi≡ξi(x,t) δa] with associated changes in trajectory parameter t defined by (c) t=t+δt [δt≡ξ0(x,t) δa]. The condition (d) δΛi(L) =0 for a symmetry mapping may be represented in the equivalent form (e) Λi(N) =0, where (f) Nδa≡δL+Ld (δt)/dt. We consider two subcases of these symmetry mappings which are referred to as R1, R2 respectively. Associated with R1 mappings [which are satisfied by a large class of Lagrangians including all L=L (χ,x)] is a time‐dependent constant of motion (g) C1≡ (∂N/∂χi) χi −N+(∂/∂t)[(∂L/∂χi) ξi−Eξ0]+γ1(x,t), where γ1 is determined by R1. The R2 subcase is the familiar Noether symmetry condition and hence has associated wi...

On symmetries, conservation laws, and variational problems for partial differential equations

The problem of the correspondence between symmetries and conservation laws for partial differential equations is considered. For Lagrangian systems the set of Noether (variational) symmetries can be shown to lead to the set of all local conservation laws. For partial differential equations without well‐defined Lagrangian functions there is no universal correspondence between symmetries and conservation laws. In this article it is shown that for a large class of differential equations there is a natural way to associate conservation laws with symmetries. The class consists of many interesting equations, e.g., Korteweg–de Vries equation, Kadomtsev–Petviashvili equation, Boussinesq equation, nonlinear diffusion equation, Monge–Ampere equation, regularized long‐wave equation, and Navier–Stokes equations. Characteristics of the corresponding conservation laws are calculated and examples are given. For Lagrangian systems the consistency of the approach with the standard Noether results is discussed.

Mechanical Conservation Laws and the Physical Properties of Groups of Motions in Flat and Curved Space-Times

The intimate relation of symmetry properties describable by groups of motions and the consequent conservation laws realizable for a particle in classical mechanics and in the mechanics of restricted and general theories of relativity are discussed in detail using elementary results of the theory of continuous groups. In addition, for the case of special relativistic mechanics the finite form of the groups of motions, underlying all of the possible constants of the motion of the form of first integrals linear in the momenta, are constructed and shown to be the inhomogeneous (i.e., including displacements) proper Lorentz transformations.

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

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 sy...

Groups of Curvature Collineations in Riemannian Space‐Times Which Admit Fields of Parallel Vectors

By definition, a Riemannian space Vn admits a symmetry called a curvature collineation (CC) if the Lie derivative with respect to some vector ξi of the Riemann curvature tensor vanishes. It is shown that if a Vn admits a parallel vector field, then it will admit groups of CCs. It follows that every space‐time with an expansion‐free, shear‐free, rotation‐free, geodesic congruence admits groups of CCs, and hence gravitational pp waves admit such groups of symmetries.

Time‐dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time‐dependent Kepler potentials

It is shown there are only two classes of time‐dependent Kepler potentials [V2≡λ0(at2+bt+c)−1/2/r, (b2−4ac≠0), and V3≡λ0(αt+β)−1/r] for which the associated classical dynamical equations will admit quadratic first integrals more general than quadratic functions of the angular momentum. In addition to the angular momentum the system defined by V2 admits only a ’’generalized time‐dependent energy integral,’’ while the system defined by V3 admits in addition to these a time‐dependent vector first integral that is a generalization of the Laplace–Runge–Lenz vector constant of motion (associated with the time‐independent Kepler system). For the V3 system the time‐dependent vector first integral is employed to obtain in a simple manner the orbit equations in completely integrated form. The complete group of (velocity‐independent) symmetry mappings is obtained for each of these two classes of dynamical systems and used to show that the generalized energy integral is expressible as a Noether constant of motion.

Related integral theorem. II. A method for obtaining quadratic constants of the motion for conservative dynamical systems admitting symmetries

By use of Lie derivatives, symmetry mappings for conservative dynamical systems are formulated in terms of continuous groups of infinitesimal transformations within the configuration space. Such symmetry transformations, called trajectory collineations, may be interpreted as point mappings which drag along coordinates and geometric objects as they map trajectories into trajectories. It is shown that if a conservative dynamical system admits a trajectory collineation, then in general a new quadratic (in the velocity) constant of the motion will result from the deformation of a given quadratic constant of the motion under such a symmetry mapping. The theory is applied to obtain the group of symmetry transformations and concomitant constants of the motion associated with the deformations of the energy integral for the Kepler problem and for the isotropic simple harmonic oscillator. The Runge‐Lenz vector of the Kepler problem and the symmetric tensor constant of the motion for the three‐dimensional oscillator...

Characteristic functional structure of infinitesimal symmetry mappings of classical dynamical systems. I. Velocity‐dependent mappings of second‐order differential equations

The primary purpose of this paper is to show that infinitesimal velocity‐dependent symmetry mappings [(a) xi =xi +δxi, δxi  ≡ ξi(x,x,t)δa with associated change in path parameter (b) t=t+δt, δt ≡ ξ0(x,x,t)] of classical (including relativistic) particle systems (c) Ei(x,x,x,t) =0 are expressible in a form with a characteristic functional structure which is the same for all dynamical systems (c) and is manifestly dependent upon constants of motion of the system. In this characteristic form the symmetry mappings are determined by (d) ξi =Zi(x,x,t) +xiξ0,ξ0 arbitrary; the functions Zi appearing in (d) have the form (e) Zi =BAgiA(C1,...,Cr; t), 0≤r≤2n, A=1,...,2n, where the BA are arbitrary constants of motion and the C’s appearing in the functions giA are specified constants of motion.A procedure is given to determine the giA. For Lagrangian systems it follows that velocity‐dependent Noether mappings are a subclass of the above‐mentioned general symmetry mappings of the form (a)–(e). An analysis of v...