scholarly journals Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation

2003 ◽  
Vol 20 (18) ◽  
pp. 4043-4066 ◽  
Author(s):  
M Ferraris ◽  
M Francaviglia ◽  
M Raiteri
Keyword(s):  
Equations Of Motion ◽  
Conserved Quantities ◽  
Theories Of Gravitation

Learning the Dynamics of Objects by Optimal Functional Interpolation

2012 ◽  
Vol 24 (9) ◽  
pp. 2457-2472
Author(s):  
Jong-Hoon Ahn ◽  
In Young Kim
Keyword(s):  
Functional Data ◽  
Continuity Equation ◽  
Particle Concentration ◽  
Equations Of Motion ◽  
Navier Stokes ◽  
Conserved Quantities ◽  
Time Varying ◽  
Science And Engineering ◽  
Navier Stokes Equation ◽  
Over Time

Many areas of science and engineering rely on functional data and their numerical analysis. The need to analyze time-varying functional data raises the general problem of interpolation, that is, how to learn a smooth time evolution from a finite number of observations. Here, we introduce optimal functional interpolation (OFI), a numerical algorithm that interpolates functional data over time. Unlike the usual interpolation or learning algorithms, the OFI algorithm obeys the continuity equation, which describes the transport of some types of conserved quantities, and its implementation shows smooth, continuous flows of quantities. Without the need to take into account equations of motion such as the Navier-Stokes equation or the diffusion equation, OFI is capable of learning the dynamics of objects such as those represented by mass, image intensity, particle concentration, heat, spectral density, and probability density.

scholarly journals Gauge-independent Theory of Symmetry. I.

1964 ◽  
Vol 17 (4) ◽  
pp. 431 ◽  
Author(s):  
LJ Tassie ◽  
HA Buchdahl
Keyword(s):  
Electromagnetic Field ◽  
Single Particle ◽  
Equations Of Motion ◽  
Classical Mechanics ◽  
Conserved Quantities ◽  
Continuous Symmetry ◽  
Symmetry Properties ◽  
Symmetry Transformations ◽  
Hamiltonian Forms

The invariance of a system under a given transformation of coordinates is usually taken to mean that its Lagrangian is invariant under that transformation. Consequently, whether or not the system is invariant will depend on the gauge used in describing the system. By defining invariance of a system to mean the invariance of its equations of motion, a gauge-independent theory of symmetry properties is obtained for classical mechanics in both the Lagrangian and Hamiltonian forms. The conserved quantities associated with continuous symmetry transformations are obtained. The system of a single particle moving in a given electromagnetic field is considered in detail for various symmetries of the electromagnetic field, and the appropriate conserved quantities are found.

scholarly journals Classical Integrability of Chiral QCD2 and Classical Curves

1997 ◽  
Vol 12 (32) ◽  
pp. 2445-2453 ◽  
Author(s):  
Robert De Mello Koch ◽  
João P. Rodrigues
Keyword(s):  
Large Class ◽  
Gauge Group ◽  
Infinite Number ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Conserved Quantities ◽  
Large N ◽  
Fermionic Fields

In this letter, classical chiral QCD 2 is studied in the lightcone gauge A-=0. The once integrated equation of motion for the current is shown to be of the Lax form, which demonstrates an infinite number of conserved quantities. Specializing to gauge group SU(2), we show that solutions to the classical equations of motion can be identified with a very large class of curves. We demonstrate this correspondence explicitly for two solutions. The classical fermionic fields associated with these currents are then obtained. Finally, we conclude by showing how 't Hooft's large-N solution is obtained from one of our solutions.

scholarly journals Hamiltonisation, measure preservation and first integrals of the multi-dimensional rubber Routh sphere

2019 ◽  
Vol 46 (1) ◽  
pp. 65-88 ◽  
Author(s):  
Luis García-Naranjo
Keyword(s):  
Invariant Measure ◽  
Mass Distribution ◽  
General Result ◽  
Equations Of Motion ◽  
First Integrals ◽  
Multiplier Method ◽  
Conserved Quantities ◽  
Hamiltonian Form ◽  
Reduced Equations ◽  
Internal Symmetries

We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from Garc?a-Naranjo [21] and Garc?a-Naranjo and Marrero [22], we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin?s reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Angular Momentum ◽  
Conservation Laws ◽  
Equations Of Motion ◽  
Geodesic Equation ◽  
First Integrals ◽  
Spacetime Symmetries ◽  
Conserved Quantities ◽  
Killing Vectors ◽  
Invariance Properties ◽  
Matter Energy

This chapter studies how the ‘spacetime symmetries’ can generate first integrals of the equations of motion which simplify their solution and also make it possible to define conserved quantities, or ‘charges’, characterizing the system. As already mentioned in the introduction to matter energy–momentum tensors in Chapter 3, the concepts of energy, momentum, and angular momentum are related to the invariance properties of the solutions of the equations of motion under spacetime translations or rotations. The chapter explores these in greater detail. It first turns to isometries and Killing vectors. The chapter then examines the first integrals of the geodesic equation, and Noether charges.

Interacting charges I

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Electromagnetic Interaction ◽  
Equations Of Motion ◽  
Inertial Mass ◽  
Dipole Approximation ◽  
Speed Of Light ◽  
Third Order ◽  
Mass System ◽  
Dipole Radiation ◽  
All Speeds ◽  
Theories Of Gravitation

This chapter addresses the problem of radiation by a system of point charges. Owing to the fact that the electromagnetic interaction propagates at finite speed, this problem can only be solved iteratively, by assuming that all speeds are small compared to the speed of light. The chapter then derives the dipole and quadrupole formulas giving the radiation field and the energy radiated by the system in the lowest orders. Finding the field and the radiation of a system of charges beyond the dipole approximation is rather more difficult, but necessary in the absence of dipole radiation. This is also a useful exercise for studying the radiation of a mass system in theories of gravitation where the gravitational mass is equal to the inertial mass. In addition, the chapter finds the equations of motion of the charges of the system to third order in the velocities.

scholarly journals GENERALIZED GALILEI-INVARIANT CLASSICAL MECHANICS

2005 ◽  
Vol 20 (18) ◽  
pp. 4259-4289
Author(s):  
HARRY W. WOODCOCK ◽  
PETER HAVAS
Keyword(s):  
Equations Of Motion ◽  
Classical Mechanics ◽  
Slow Motion ◽  
Multiple Time ◽  
Conserved Quantities ◽  
Interacting Particles ◽  
Initial Value ◽  
Slow Motions ◽  
Advanced Interaction ◽  
Delay Differential

To describe the "slow" motions of n interacting mass points, we give the most general four-dimensional (4D) noninstantaneous, nonparticle symmetric Galilei-invariant variational principle. It involves two-body invariants constructed from particle 4-positions and 4-velocities of the proper orthochronous inhomogeneous Galilei group. The resulting 4D equations of motion and multiple-time conserved quantities involve integrals over the worldlines of the other n-1 interacting particles. For a particular time-asymmetric retarded (advanced) interaction, we show the vanishing of all integrals over worldlines in the ten standard 4D multiple-time conserved quantities, thus yielding a Newtonian-like initial value problem. This interaction gives 3D noninstantaneous, nonparticle symmetric, coupled nonlinear second-order delay-differential equations of motion that involve only algebraic combinations of nonsimultaneous particle positions, velocities, and accelerations. The ten 3D noninstantaneous, nonparticle symmetric conserved quantities involve only algebraic combinations of nonsimultaneous particle positions and velocities. A two-body example with a generalized Newtonian gravity is provided. We suggest that this formalism might be useful as an alternative slow-motion mechanics for astrophysical applications.

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