Related papers: The Generalized Jacobi Equation
By using the relativistic top theory, we derive a relativistic top deviation equation. This equation turns out to be a generalization of the geodesic deviation equation for a pair of nearby point particles. In fact, we show that when the…
We discuss some properties of Jacobi fields that do not involve assumptions on the curvature endomorphism. We compare indices of different spaces of Jacobi fields and give some applications to Riemannian geometry.
A nonrelativistic particle released from rest at the edge of a ball of uniform charge density or mass density oscillates with simple harmonic motion. We consider the relativistic generalizations of these situations where the particle can…
We derive generalized geodesic equations in curved spacetime that include conservative forces, dissipative effects, and quantum-gravity-motivated minimal-length corrections. Conservative interactions are incorporated through external vector…
The geometric formulation of Hamilton--Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton--Jacobi problem with the…
Geodesic deviation is the most basic manifestation of the influence of gravitational fields on matter. We investigate geodesic deviation within the framework of Regge calculus, and compare the results with the continuous formulation of…
The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of…
The long-time behavior of stochastic Hamilton-Jacobi equations is analyzed, including the stochastic mean curvature flow as a special case. In a variety of settings, new and sharpened results are obtained. Among them are (i) a…
In most text books of mechanics, Newton's laws or Hamilton's equations of motion are first written down and then solved based on initial conditions to determine the constants of the motions and to describe the trajectories of the particles.…
For an investigation of the physical properties of gravitational fields the observation of massive test particles and light is very useful. The characteristic features of a given space-time may be decoded by studying the complete set of all…
The present work investigates the evolution of linear perturbations of time-dependent ideal fluid flows with advected quantities, expressed in terms of the second order variations of the action corresponding to a Lagrangian defined on a…
The geometric approach to optimal transport and information theory has triggered the interpretation of probability densities as an infinite-dimensional Riemannian manifold. The most studied Riemannian structures are Otto's metric, yielding…
In this paper, we give a rigorous derivation of Einstein's geodesic hypothesis in general relativity. We use scaling stable solitons for nonlinear wave equations to approximate the test particle. Given a vacuum spacetime $([0,…
The evolution equations of Einstein's theory and of Maxwell's theory---the latter used as a simple model to illustrate the former--- are written in gauge covariant first order symmetric hyperbolic form with only physically natural…
The Hamilton-Jacobi equation of classical mechanics is approached as a model reduction of conservative particle mechanics where the velocity degrees-of-freedom are eliminated. This viewpoint allows an extension of the association of the…
The geodesic flow on the tangent bundle is the flow of a certain vector field which is called the spray $S:TM\to TTM$. The flow lines of the vector field $\ka_{TM}\o TS:TTM\to TTTM$ project to the Jacobi fields on $TM$. This could be called…
We generalize the Hamilton-Jacobi formulation for higher order singular systems and obtain the equations of motion as total differential equations. To do this we first study the constraint structure present in such systems.
A detailed Hamilton-Jacobi analysis for linearized $\lambda R$ gravity is developed. The model is constructed by rewriting linearized gravity in terms of a parameter $\lambda$ and new variables. The set of all hamiltonians is identified…
The article surveys inverse problems related to the twisted geodesic flows on Riemannian manifolds with boundary, focusing on the generalized ray transforms, tensor tomography, and rigidity problems. The twisted geodesic flow generalizes…
We show that gravitational waves can act as waveguides for electromagnetic radiation, that is if the latter is initially aligned with the gravitational waves, then the alignment will survive during the propagation. The analysis is performed…