Gravitational collapse and higher-order gravitational lagrangians

A general form of higher-order contributions (in R_ij) to the Einstein field equations is displayed. The additional terms may either stabilize or destabilize self-gravitating objects in gravitational collapse depending on the sign of the coefficient introducing the quadratic term. If the quadratic term is stabilizing, intertial mass can be converted to radiation with an efficiency approaching 100%, and arbitrarily large masses can be stabilized. On the other hand, the resultant field equations are pathological in that they admit gravitons with negative mass-squared (i.e., tachyons). A nonsingular class of vacuum solutions exist in general for the quadratic case (“grey dimples”).     

Cosmological singularities and higher-order gravitational lagrangians

General relativity is modified by adding terms proportional to R² and R^μνR_μν to the Lagrangian. One class of solutions of the modified field equations is free of singularities but does not lead to asymptotic behaviour (for large time) of the Friedmann type. A second class, which shows the correct asymptotic behaviour, does contain the usual singularities of Friedmann universes, collapse being modulated by small oscillations only. The quantum effects considered here are thus unable to prevent the occurrence of cosmological singularities under physically reasonable conditions.     

Noether theorem in higher-order Lagrangian mechanics

We study higher-order Lagrangian mechanics on thek-velocity manifold. The variational problem gives rise to new concepts, such as main invariants, Zermelo conditions, higher-order energies, and new conservation laws. A theorem of Noether type is proved for higher-order Lagrangians. The invariants to the infinitesimal symmetries are explicitly written. All this construction is a natural extension of classical Lagrangian mechanics.     

Symmetries,integrating factors and Nambu mechanics

We show that if an n-dimensional autonomous dynamical system (DS) with a vector field (VF) which has constant divergence possesses n − 1 independent first integrals, then it admits a symmetry VF which involves Nambu mechanics (NM). If the DS is conservative, then the Nambu VF happens to be a symmetry VF of the DS. We also show that the integrating factors can be constructed via NM. We illustrate our results on the Lotka-Volterra DS.     

Lagrangian submanifolds and the Euler-Lagrange equations in higher-order mechanics

A direct construction of the Euler-Lagrange equations in higher-order mechanics as a submanifold of a higher-order tangent bundle is given, starting from the Lagrangian submanifold defined by the Lagrangian function. This construction uses higher-order tangent bundle geometry, derives the Euler-Lagrange equations as the constraint equations of a submanifold, and makes no assumptions about the regularity of the Lagrangian.     

Symmetries of massless Dirac equations and supersymmetric quantum mechanics

It is shown how symmetries of Dirac equations can be used to obtain constants of motion for nonrelativistic supersymmetric quantum Hamiltonians. In particular, conserved supercharges are found for a spin-1/2 particle in the field of a dyon which yield under anticommutation the generalized Runge-Lenz symmetry of the system.This work is supported in part through funds provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada and the Fonds FCAR of the Quebec Ministry of Education.     

Recursion operators, higher-order symmetries and superintegrability in quantum mechanics

A connection between the theory of superintegrable quantum-mechanical systems, which admit a maximal number of integrals of motion, and the standard Lie group theory is established. It is shown that the flows generated by first- and second-order Lie symmetries of the bidimensional Schrödinger equation can be classified and interpreted as quantum-mechanical operators which commute with integrable or superintegrable Hamiltonians. In this way, all known superintegrable potentials in the plane are naturally obtained and slightly more general integrals of motion are found.     

广义经典力学系统的对称性与Mei守恒量

研究广义经典力学系统的对称性和一类新型守恒量——Mei守恒量.在高维增广相空间中建立 了系统的运动微分方程；给出了系统的Mei对称性、Noether对称性和Lie对称性的判据；得 到了分别由三种对称性导致Mei守恒量的条件和Mei守恒量的形式.举例说明结果的应用. 关键词： 广义经典力学 Mei对称性 Noether对称性 Lie对称性 守恒量     

Acceleration-dependent lagrangians in general relativity

In contrast to electromagnetic theory, in general relativity the elimination of acceleration terms in a lagrangian by substituting into the lagrangian the equations of motion which follow from the lagrangian is a correct procedure; it corresponds to a gauge transformation.     

Nonlinear lagrangians and Einstein spaces

It is shown that, for a Riemannian spaceV _d of dimensiond, solutions of the equation ((–g)^1/2 R ⁿ)/g^ab = 0 forn = (1/4)(d+2) may be interpreted as (d + 1)-dimensional Einstein spaces.     

On quadratic lagrangians in General Relativity

Theories of gravitation similar to General Relativity but with an additionalR ² term in the Lagrangian are explored. The Schwarzschild metric is not the exterior solution that can be continued to the interior of the body to give a positive definite mass distribution. The experimental consequences ofR ² terms are investigated. Furthermore, it is shown that a theory with anR ² term only possesses an interesting singular dependence on the coupling constant.     

String loop divergences and effective lagrangians

We isolate logarithmic divergences from bosonic string amplitudes on a disc. These divergences are compared with “tadpole” divergences in the effective field theory, with a covariant cosmological term implied by the counting of string coupling constants. We find an inconsistency between the two. This might be a problem in eliminating divergences from the bosonic string.     

Acceleration-dependent lagrangians and equations of motion

Acceleration terms in a lagrangian are sometimes eliminated by substituting into the lagrangian the equations of motion which were obtained from the lagrangian. We show that, in general this is an incorrect procedure. In addition, we present a new correct procedure, which we call the method of the double zero.     

Non-minimal chiral anomalies and effective lagrangians

Minimality of the chiral anomaly is discussed. It is shown that the consequences of low energy theorems in effective theories involving Goldstone bosons are determined solely by the equivalence class of the anomaly and not by the representative chosen.     

An alternative to wave mechanics on curved spaces

Geodesic motion in infinite spaces of constant negative curvature provides for the first time an example where a basically quantum mechanical quantity, a ground-state energy, is derived from Newtonian mechanics in a rigorous, nonsemiclassical way. The ground state energy emerges as the Hausdorff dimension of a quasi-self-similar curve at infinity of three-dimensional hyperbolic spaceH ³ in which our manifolds are embedded and where their universal covers are realized. This curve is just the locus of the limit set () of the Kleinian group of covering transformations, which determines the bounded trajectories in the manifold; all of them lie in the quotientC(), C() being the hyperbolic convex hull of (). The three-dimensional hyperbolic manifolds we construct can be visualized as thickened surfaces, topological productsI×S, I a finite open interval, the fibersS compact Riemann surfaces. We give a short derivation of the Patterson formula connecting the ground-state energy with the Hausdorff dimension of , and give various examples for the calculation of from the tessellations of the boundary ofH ³, induced by the universal coverings of the manifolds.     

Internal symmetry currents and supersymmetric effective lagrangians

A supersymmetric and non-linearly realized internally symmetric action is constructed from the super Kähler potential of Goldstone scalar superfields. Noether's theorem in superspace is derived and the associated superfield of currents defined. The currents are used to derive a super Dashen formula relating the (quasi) Goldstone masses and decay constants to the symmetry breaking part of the theory and to supersymmetrically gauge the invariant subgroup as well as the full group.     

Anomalies and effective lagrangians of superconducting cosmic strings

We study the effective lagrangians of superconducting cosmic strings and the underlying anomaly structure leading to various effects. We show that fermionic superconducting cosmic (Nielsen-Olesen) strings arise as a consequence of a mixed anomaly of the form E · B′ where E is the electromagnetic field strength and B′ the core flux of the string. We show how this controls superconductivity, the index theorems, topological mass terms on domain walls, and string-axion interactions. We contrast this to global strinfs with Chern-Simons currents.