Lagrangian equations of motion. Conservation laws

2020 ◽  
pp. 156-174
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Conservation Laws ◽  
Coordinate Transformation ◽  
Virial Theorem ◽  
Equations Of Motion ◽  
Electromechanical System ◽  
Interacting Particles ◽  
Integral Of Motion ◽  
Lagrangian Functions ◽  
Lagrangian Equations

This chapter addresses the invariance of the Lagrangian equations of motion under the coordinate to transformation, the transformation of the energy and generalised momenta under the coordinate transformation. The integrals of motion for a particle moving in the field with a given symmetry to the Noether’s theorem, the Lagrangian functions, and the Lagrangian equations of motion for the electromechanical system. The authors also discuss the influence of constraints and friction on the motion of a system, the virial theorem and its generalization in the presents of a magnetic field, and an additional integral of motion for a system of three interacting particles.

Lagrangian equations of motion. Conservation laws

2020 ◽  
pp. 16-24
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Conservation Laws ◽  
Coordinate Transformation ◽  
Virial Theorem ◽  
Equations Of Motion ◽  
Electromechanical System ◽  
Interacting Particles ◽  
Integral Of Motion ◽  
Lagrangian Functions ◽  
Lagrangian Equations

This chapter addresses the invariance of the Lagrangian equations of motion under the coordinate to transformation, the transformation of the energy and generalised momenta under the coordinate transformation. The integrals of motion for a particle moving in the field with a given symmetry to the Noether’s theorem, the Lagrangian functions, and the Lagrangian equations of motion for the electromechanical system. The authors also discuss the influence of constraints and friction on the motion of a system, the virial theorem and its generalization in the presents of a magnetic field, and an additional integral of motion for a system of three interacting particles.

Lagrangian equations of motion for a massive conductor in a magnetic field

1994 ◽  
Vol 37 (7) ◽  
pp. 651-656
Author(s):  
Yu. G. Pavlenko ◽  
Yu. M. Petrov
Keyword(s):  
Magnetic Field ◽  
Equations Of Motion ◽  
Lagrangian Equations

A Strategy to Accelerate the Numerical Integration in the Dynamic Simulation of Multibody Systems

1997 ◽  
Author(s):  
Jesús Cardenal ◽  
Javier Cuadrado ◽  
Eduardo Bayo
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Stiff Systems ◽  
Step Size ◽  
Step Method ◽  
Integral Of Motion ◽  
Time Step ◽  
Time Step Size ◽  
Variable Time ◽  
Numerical Integrator

Abstract This paper presents a multi-index variable time step method for the integration of the equations of motion of constrained multibody systems in descriptor form. The basis of the method is the augmented Lagrangian formulation with projections in index-3 and index-1. The method takes advantage of the better performance of the index-3 formulation for large time steps and of the stability of the index-1 for low time steps, and automatically switches from one method to the other depending on the required accuracy and values of the time step. The variable time stepping is accomplished through the use of an integral of motion, which in the case of conservative systems becomes the total energy. The error introduced by the numerical integrator in the integral of motion during consecutive time steps provides a good measure of the local integration error, and permits a simple and reliable strategy for varying the time step. Overall, the method is efficient and powerful; it is suitable for stiff and non-stiff systems, robust for all time step sizes, and it works for singular configurations, redundant constraints and topology changes. Also, the constraints in positions, velocities and accelerations are satisfied during the simulation process. The method is robust in the sense that becomes more accurate as the time step size decreases.

Dynamics of Spinning Disc Parametrically Excited by Noise

1999 ◽  
Author(s):  
Narayanan Ramakrishnan ◽  
N. Sri Namachchivaya
Keyword(s):  
Equations Of Motion ◽  
Planck Equation ◽  
Stochastic Averaging ◽  
Transverse Motion ◽  
Probability Density Distribution ◽  
Integral Of Motion ◽  
Parametrically Excited ◽  
One Dimensional ◽  
Small Intensity ◽  
Spinning Disc

Abstract The nonlinear dynamics of a circular spinning disc parametrically excited by noise of small intensity is investigated. The governing PDEs are reduced using a Galerkin reduction procedure to a two-DOF system of ODEs which, govern the transverse motion of the disc. The dynamics is simplified by exploiting the S1 invariance of the equations of motion of the reduced system and further, reduced by performing stochastic averaging. The resulting one-dimensional Markov diffusive process is studied in detail. The stationary probability density distribution is obtained by solving the Fokker-Planck equation along with the appropriate boundary conditions. The boundary behaviour is studied using an asymptotic approach. Some aspects of dynamical and phenomenological bifurcations of the stationary solution are also investigated. The scheme of things presented here can be applied in principle to a four-dimensional Hamiltonian system possessing one integral of motion in addition to the hamiltonian and having one fixed point.

Lagrangian Mechanics

2019 ◽  
pp. 53-82
Author(s):  
David D. Nolte
Keyword(s):  
Dynamical Systems ◽  
Conservation Laws ◽  
Euler Equations ◽  
Virial Theorem ◽  
Variational Calculus ◽  
Mechanical Action ◽  
Lagrange Equations ◽  
Time Average ◽  
The Difference ◽  
Kinetic And Potential Energies

Dynamical systems follow trajectories for which the mechanical action integrated along the trajectory is an extremum. The action is defined as the time average of the difference between kinetic and potential energies, which is also the time average of the Lagrangian. Once a Lagrangian has been defined for a system, the Euler equations of variational calculus lead to the Euler–Lagrange equations of dynamics. This chapter explores applications of Lagrangians and the use of Lagrange’s undetermined multipliers. Conservation laws, central forces, and the virial theorem are developed and explained.

scholarly journals Dynamics of an orbital polarization of twisted electron beams in electric and magnetic fields

2019 ◽  
Vol 204 ◽  
pp. 10008
Author(s):  
Alexander J. Silenko ◽  
Pengming Zhang ◽  
Liping Zou
Keyword(s):  
Magnetic Field ◽  
Electron Beam ◽  
Magnetic Fields ◽  
Quantum Dynamics ◽  
Electron Beams ◽  
Equations Of Motion ◽  
Vortex Beam ◽  
Dirac Particles ◽  
Electric And Magnetic Fields ◽  
Average Projection

Relativistic classical and quantum dynamics of twisted (vortex) Dirac particles in arbitrary electric and magnetic fields is constructed. The relativistic Hamiltonian and equations of motion in the Foldy-Wouthuysen representation are derived. Methods for the extraction of an electron vortex beam with a given orbital polarization and for the manipulation of such a beam are developed. The new effect of a radiative orbital polarization of a twisted electron beam in a magnetic field resulting in a nonzero average projection of the intrinsic orbital angular momentum on the field direction is predicted.

scholarly journals 50. Magneto-hydrostatic equilibrium in an external magnetic field

1958 ◽  
Vol 6 ◽  
pp. 499-503 ◽  
Author(s):  
P. A. Sweet
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Electric Current ◽  
Virial Theorem ◽  
Magnetic Energy ◽  
Hydrostatic Equilibrium

The expression ∫∫∫all space ΔH2dv for a change in magnetic energy is shown to be incorrect when applied to a body carrying an electric current and situated in an external magnetic field. A modified expression is derived.Chandrasekhar's form of the virial theorem in a magnetic field is extended to the case where an external magnetic field is present.

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