Solutions of the d'Alembert equation in ∝n

1975 ◽  
pp. 14-38
Author(s):  
Calvin H. Wilcox
Keyword(s):  
D’Alembert Equation

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon
Keyword(s):  
Symmetry Group ◽  
Mechanical Systems ◽  
Hamiltonian Formalism ◽  
Geometrical Approach ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Relative Configuration ◽  
D’Alembert Equation ◽  
Dynamical Reduction ◽  
Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

scholarly journals Functional Equations and Fourier Analysis

2013 ◽  
Vol 56 (1) ◽  
pp. 218-224 ◽  
Author(s):  
Dilian Yang
Keyword(s):  
Representation Theory ◽  
Fourier Analysis ◽  
Harmonic Analysis ◽  
Functional Equations ◽  
Compact Groups ◽  
Wilson Equation ◽  
D’Alembert Equation

AbstractBy exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation-on compact groups.

On weak solutions to the Lagrange–d'Alembert equation

2013 ◽  
Vol 40 (3) ◽  
pp. 383-392
Author(s):  
Dmitry Treschev ◽  
Oleg Zubelevich
Keyword(s):  
Weak Solutions ◽  
D’Alembert Equation

scholarly journals On solutions of the d’Alembert equation on a restricted domain

2012 ◽  
Vol 85 (1-2) ◽  
pp. 169-183 ◽  
Author(s):  
Anna Bahyrycz ◽  
Janusz Brzdȩk
Keyword(s):  
Restricted Domain ◽  
D’Alembert Equation

A DIRECT DISCRETE FORMULATION FOR THE WAVE EQUATION

2001 ◽  
Vol 09 (04) ◽  
pp. 1355-1382 ◽  
Author(s):  
ENZO TONTI
Keyword(s):  
Wave Equation ◽  
Close Link ◽  
Cell Complexes ◽  
Global Variables ◽  
Differential Formulation ◽  
D’Alembert Equation ◽  
Physical Laws ◽  
Discrete Formulation ◽  
Preliminary Classification

The paper shows how to give a direct discrete formulation of the wave equation starting directly from physical laws, i.e. without passing through differential formulation. Using global variables instead of scalar and vector field functions, a close link between global variables and spatial and temporal elements immediately appears. A preliminary classification of physical variables into three classes: configuration, source and energy variables and the use of two cell complexes, one dual of the other, gives an unambiguous association of global variables to the spatial and temporal elements of the two complexes. Thus, one arrives at a discrete formulation of d'Alembert equation on an unstructured mesh.

On hidden symmetries and solutions of the nonlinear d’Alembert equation

2013 ◽  
Vol 18 (7) ◽  
pp. 1589-1599 ◽  
Author(s):  
Anatoliy F. Barannyk ◽  
Tetyana A. Barannyk ◽  
Ivan I. Yuryk
Keyword(s):  
Hidden Symmetries ◽  
D’Alembert Equation

A simple method of finding solutions of the nonlinear d'Alembert equation

1992 ◽  
Vol 25 (14) ◽  
pp. L871-L877 ◽  
Author(s):  
P Basarab-Horwath ◽  
W Fushchich ◽  
M Serov
Keyword(s):  
Simple Method ◽  
D’Alembert Equation
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