dynamics of a particle: free_motion

Dynamics of a Particle: Free Motion

The differential equations for the motion of a particle under any forces when we use rectangular coordinates are known to be

X,Y, and Z, the components of the actual forces on the particle resolved parallel to the fixed rectangular axes, or rather their equivalents mẍ,mÿ,mz, are called the effective forces on the particle. They are of course a set of forces mechanically equivalent to the actual forces acting on the particle.

The equations of motion of the particle in terms of any other system of coordinates are easily obtained.

Let q₁,q₂,q₃ be the coordinates in question. The appropriate formulas for transformation of coordinates express x,y, and z in terms of q₁,q₂, and q₃:

For the component velocity ẋ we have

and ẋ,ẏ,ż are explicit functions of q₁,q₂,q₃,q₁,q₂,q₃, linear and homogeneous in terms of q₁,q₂,q₃.¹

We may note in passing that it follows from this fact that ẋ²,ẏ², and ż² are homogeneous quadratic functions of q₁,q₂, and q₃.

Obviously

and since

and

we have

Let us find now an expression for the work δ_q1W done by the effective forces when the coordinate q₁ is changed by an infinitesimal amount δq₁ without changing q₂ or q₃. If δx,δy,δz are the changes thus produced in x,y,z, obviously

If expressed in rectangular coordinates, we need, however, to express δ_q1W in terms of our coordinates q₁,q₂,q₃:

Now

but by (2) and (3),

Hence

Therefore

where

is the kinetic energy of the particle.

To get our differential equation we have only to write the second member of (4) equal to the work done by the actual forces when q₁ is changed by δq₁.

If we represent the work in question by Q₁δq₁, our equation is

and of course we get such an equation for every coordinate.

It must be noted that usually equation (5) will contain q₂ and q₃ and their time derivatives as well as q₁, and therefore cannot be solved without the aid of the other equations of the set.

In any concrete problem, T must be expressed in terms of q₁,q₂,q₃ and their time derivatives before we can form the expression for the work done by the effective forces. The work done by the actual forces must be obtained from direct examination of the problem.

Source

William Elwood Byerly, An Introduction to the Use of Generalized Coördinates in mechanics and Physics, Ginn and Company, 1916. Chapter I, “Introduction.”

The 1916 source work is in the public domain in the United States.