scholarly journals THE FORCE OF GRAVITY IN SCHWARZSCHILD AND GULLSTRAND–PAINLEVÉ COORDINATES

2009 ◽  
Vol 18 (14) ◽  
pp. 2289-2294 ◽  
Author(s):  
CARL BRANNEN
Keyword(s):  
Equations Of Motion ◽  
Relativistic Correction ◽  
First Order ◽  
Geodesic Equations ◽  
Affine Parameter ◽  
Tests Of Gravity

We derive the exact equations of motion (in Newtonian, F = ma, form) for test masses in Schwarzschild and Gullstrand–Painlevé coordinates. These equations of motion are simpler than the usual geodesic equations obtained from Christoffel tensors, in that the affine parameter is eliminated. The various terms can be compared against tests of gravity. In force form, gravity can be interpreted as resulting from a flux of superluminal particles (gravitons). We show that the first order relativistic correction to Newton's gravity results from a two-graviton interaction.

scholarly journals DIRAC ANALYSIS AND INTEGRABILITY OF GEODESIC EQUATIONS FOR CYLINDRICALLY SYMMETRIC SPACETIMES

2003 ◽  
Vol 12 (08) ◽  
pp. 1431-1444 ◽  
Author(s):  
UGUR CAMCI
Keyword(s):  
Symplectic Structure ◽  
Equations Of Motion ◽  
First Order ◽  
Geodesic Equations ◽  
Cylindrically Symmetric ◽  
Symmetric Spacetimes ◽  
Constraint Analysis ◽  
Stationary Spacetime ◽  
Cylindrically Symmetric Spacetimes ◽  
Geodesic Equations Of Motion

Dirac's constraint analysis and the symplectic structure of geodesic equations are obtained for the general cylindrically symmetric stationary spacetime. For this metric, using the obtained first order Lagrangian, the geodesic equations of motion are integrated, and found some solutions for Lewis, Levi-Civita, and Van Stockum spacetimes.

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

Analysis of Linear Nonconservative Vibrations

1995 ◽  
Vol 62 (3) ◽  
pp. 685-691 ◽  
Author(s):  
F. Ma ◽  
T. K. Caughey
Keyword(s):  
Modal Analysis ◽  
Equations Of Motion ◽  
Substantial Reduction ◽  
Computational Effort ◽  
Equivalence Transformations ◽  
State Space Approach ◽  
Nonconservative System ◽  
First Order ◽  
Physical Insight ◽  
Space Approach

The coefficients of a linear nonconservative system are arbitrary matrices lacking the usual properties of symmetry and definiteness. Classical modal analysis is extended in this paper so as to apply to systems with nonsymmetric coefficients. The extension utilizes equivalence transformations and does not require conversion of the equations of motion to first-order forms. Compared with the state-space approach, the generalized modal analysis can offer substantial reduction in computational effort and ample physical insight.

scholarly journals Actions, topological terms and boundaries in first-order gravity: A review

2016 ◽  
Vol 25 (04) ◽  
pp. 1630011 ◽  
Author(s):  
Alejandro Corichi ◽  
Irais Rubalcava-García ◽  
Tatjana Vukašinac
Keyword(s):  
Boundary Conditions ◽  
Symplectic Structure ◽  
Equations Of Motion ◽  
Hamiltonian Formalism ◽  
Action Principle ◽  
Internal Boundary ◽  
Four Dimensions ◽  
First Order ◽  
Conserved Charges ◽  
Boundary Terms

In this review, we consider first-order gravity in four dimensions. In particular, we focus our attention in formulations where the fundamental variables are a tetrad [Formula: see text] and a [Formula: see text] connection [Formula: see text]. We study the most general action principle compatible with diffeomorphism invariance. This implies, in particular, considering besides the standard Einstein–Hilbert–Palatini term, other terms that either do not change the equations of motion, or are topological in nature. Having a well defined action principle sometimes involves the need for additional boundary terms, whose detailed form may depend on the particular boundary conditions at hand. In this work, we consider spacetimes that include a boundary at infinity, satisfying asymptotically flat boundary conditions and/or an internal boundary satisfying isolated horizons boundary conditions. We focus on the covariant Hamiltonian formalism where the phase space [Formula: see text] is given by solutions to the equations of motion. For each of the possible terms contributing to the action, we consider the well-posedness of the action, its finiteness, the contribution to the symplectic structure, and the Hamiltonian and Noether charges. For the chosen boundary conditions, standard boundary terms warrant a well posed theory. Furthermore, the boundary and topological terms do not contribute to the symplectic structure, nor the Hamiltonian conserved charges. The Noether conserved charges, on the other hand, do depend on such additional terms. The aim of this manuscript is to present a comprehensive and self-contained treatment of the subject, so the style is somewhat pedagogical. Furthermore, along the way, we point out and clarify some issues that have not been clearly understood in the literature.

Consistent section-averaged equations of quasi-one-dimensional laminar flow

2010 ◽  
Vol 656 ◽  
pp. 337-341 ◽  
Author(s):  
PAOLO LUCHINI ◽  
FRANÇOIS CHARRU
Keyword(s):  
Equations Of Motion ◽  
Stability Result ◽  
Momentum Equation ◽  
Dissipation Function ◽  
Dimensional Solution ◽  
First Order ◽  
Averaged Equations ◽  
Classical Stability ◽  
Momentum Integral ◽  
Free Falling

Section-averaged equations of motion, widely adopted for slowly varying flows in pipes, channels and thin films, are usually derived from the momentum integral on a heuristic basis, although this formulation is affected by known inconsistencies. We show that starting from the energy rather than the momentum equation makes it become consistent to first order in the slowness parameter, giving the same results that have been provided until today only by a much more laborious two-dimensional solution. The kinetic-energy equation correctly provides the pressure gradient because with a suitable normalization the first-order correction to the dissipation function is identically zero. The momentum equation then correctly provides the wall shear stress. As an example, the classical stability result for a free falling liquid film is recovered straightforwardly.

Dynamic Analysis of Elastic Beam-Like Mechanism Systems

1989 ◽  
Vol 111 (4) ◽  
pp. 626-629
Author(s):  
W. Ying ◽  
R. L. Huston
Keyword(s):  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Cantilever Beam ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Order Perturbation ◽  
First Order ◽  
Geometric Stiffness ◽  
Stiffness Matrices ◽  
First Order Perturbation

In this paper the dynamic behavior of beam-like mechanism systems is investigated. The elastic beam is modeled by finite rigid segments connected by joint springs and dampers. The equations of motion are derived using Kane’s equations. The nonlinear terms are linearized by first order perturbation about a system balanced configuration state leading to geometric stiffness matrices. A simple numerical example of a rotating cantilever beam is presented.

scholarly journals ANISOTROPIC NULL STRING COSMOLOGIES

1998 ◽  
Vol 13 (39) ◽  
pp. 3169-3177 ◽  
Author(s):  
IOANNIS GIANNAKIS ◽  
K. KLEIDIS ◽  
A. KUIROUKIDIS ◽  
D. PAPADOPOULOS
Keyword(s):  
Equations Of Motion ◽  
String Tension ◽  
Perturbative Expansion ◽  
Speed Of Light ◽  
Zeroth Order ◽  
First Order ◽  
Cosmological Background ◽  
String Propagation ◽  
Analytical Expressions ◽  
Null String

We study string propagation in an anisotropic, cosmological background. We solve the equations of motion and the constraints by performing a perturbative expansion of the string coordinates in powers if c2 — the worldsheet speed of light. To zeroth order the string is approximated by a tensionless string (since c is proportional to the string tension T). We obtain exact, analytical expressions for the zeroth- and first-order solutions and we discuss some cosmological implications.

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

A Large Displacement Formulation Using Euler’s Angles

1999 ◽  
Author(s):  
G. Biakeu ◽  
F. Thouverez ◽  
J. P. Laine ◽  
L. Jezequel
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Static Solution ◽  
Element Formulation ◽  
First Order ◽  
Nonlinear Relation ◽  
Newton Raphson ◽  
Body Formulation ◽  
Multi Body ◽  
Euler’S Angles

Abstract The goal of this paper is to present a flexible multi-body formulation involving large displacements. This method is based on a separate discretisation of the kinetic and the internal energies. To introduce flexibility, we discretize the structure in elements (of two nodes): on each element of the beam discretisation, the local frame is defined using Euler’s angles. A finite element formulation is then applied to describe the evolution of these angles along the beam neutral fibre. For the kinetic energy, each element is cut into two rigid bars whose characteristics are given by a first order Taylor factorisation on the general kinetic energy expression. These bars are linked by a nonlinear relation. We obtain the equations of motion by applying the Lagrange’s equations to the system. These equations are solved using the Newmark method in dynamic and a Newton-Raphson technique while looking for a static solution. The method is then applied to very classic problems such as the curved beam problem proposed by authors such as Simo [6, 9], Lee [4] or the rotational rod presented by Avello [1] and Simo [7, 8] etc...

Optimal systems and their group-invariant solutions to geodesic equations

2019 ◽  
Vol 16 (09) ◽  
pp. 1950135
Author(s):  
Bismah Jamil ◽  
Tooba Feroze ◽  
Muhammad Safdar
Keyword(s):  
Nonlinear Systems ◽  
Separation Of Variables ◽  
Noether Symmetries ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
First Order ◽  
Geodesic Equations ◽  
Integrating Factor ◽  
Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

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