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Simulation of Forces : Solution of the Inverse Dynamics in Classical Mechanics


Robert Kragler
FH Ravensburg-Weingarten /University of Applied Sciences
kragler@fh-weingarten.de
A.N. Prokopenya and N.I. Chopchits
Brest Polytechnic Institute, Belarus (box@aprokop.belpak.brest.by)

1999 Mathematica Developer Conference
Applications of Mathematica
October 23, 1999


Initializations

Motivation

In general, studying the equations of motion of any physical system it is necessary to solve differential equations. However, very seldomly the corresponding solutions can be found in analytical form. Thus, usually only systems which are simple enough will be analysed in a physics course at universities. Situation changed considerably, however, with the availability of computer algebra systems. Using Mathematica in many cases we can easily find either analytical or numerical solutions of the equations of motion and visualize them even in the case of an intricate physical system. The analysis of such systems therefore promotes better understanding of the underlying physical principles and may help to develop physical intuition of the students.

Generation of Figures

The Problem of Simulation of Forces

In classical mechanics the motion of a particle is determined by Newton's second law

[Graphics:Images/index_gr_38.gif]

where the position of the particle [Graphics:Images/index_gr_39.gif] is given in Cartesian coordinates.
Here it is not assumed that this expression is a definition of the force [Graphics:Images/index_gr_40.gif] as it is e.g. in the case of [Graphics:Images/index_gr_41.gif], defining the momentum [Graphics:Images/index_gr_42.gif] of the particle, because forces can be determined independently on the lhs of Newton's second law.

Traditionally all forces considered in classical mechanics depend on coordinates and velocity of the particle, [Graphics:Images/index_gr_43.gif] , and henceforth are well known. If, however, the particle interacts with another body whose law of motion is given then the force [Graphics:Images/index_gr_44.gif] may even explicitly depend on time. Thus, in the general case the force is a function of three variables : [Graphics:Images/index_gr_45.gif] and Newton's second law combined with initial conditions completely determines the motion of the particle.

On the other hand, if the law of motion of the particle [Graphics:Images/index_gr_46.gif] is given then the resulting force [Graphics:Images/index_gr_47.gif] acting on the particle can be found as

[Graphics:Images/index_gr_48.gif]

and may be presented as a function of time [Graphics:Images/index_gr_49.gif] or as a function of velocity [Graphics:Images/index_gr_50.gif], of coordinates or some combinations of these variables.

The very first example will demonstrate that simulation of forces which occur in the equations of motion is not a trivial problem.

Example 1 : Constant Force for Particle Moving along a Straight Line
Example 2 : Particle on a Track
Example 3 : Particle on a Rotating Contour