The Equations of Motion

© 2003 David Barwacz

4/06/2003

daveb@triton.net

http://members.triton.net/daveb

Abstract

Using the space/time geometry developed in the previous paper (“Linear Motion in Space-Time, the Dirac Matrices, and Relativistic Quantum Mechanics” ), I shall show clearly why the Lagrangian equation has been a successful tool in both classical, relativistic and quantum physics.

A general form of the energy equation.

We start with Figure 1 below in which the energy triangle previously discussed, has been drawn and each segment labeled.

Substituting for the term, yields Figure 2 below.

In the remainder of this paper the space/time velocity shall simply be designated as .

We define U and T as follows.

It is clear that the total energy is simply given by .

If the total energy is constant () then, m is a constant.

Each component of the total energy as well as the total energy can be written in the general functional form:

(1.10)

Furthermore, we know the classical energy equation is also in the form of 1.10.

Consider a point object which has a spatial position variable . Clearly there is an associated velocity in the q direction given by .

Both and can be written as functions of time t.

(1.20)

(1.30)

Taking the inverse function of 1.20 and substituting it into 1.30 yields as a function of .

Define a function (1.40)

Clearly 1.40 is in the form of 1.10.

It is easy to see that (1.50)

Also, taking to be a function of , we can see that (1.60)

Now,

We can therefore conclude that = (1.70)

1.70 is the Lagrangian equation of motion. There is a discrepancy between the above derivation and the Lagrangian methods used in physics today. In the above formulization, was not independent of . In fact, it was necessary to make a function of in order to arrive at the Lagrangian equation. Next we shall overcome this problem.

Consider the function: (1.80)

Clearly 1.80 is in the form of the general energy equation 1.10.

We consider the second term to be a function of by using the substitution to arrive at: (1.90)

Considering and to be independent variables, we see that:

But and hence

We can therefore conclude that for a function of the form of 1.90 where and are treated as independent variables.

We now have the necessary background to formulate and actual physical Lagrangian.

We know that .

Consider the following equation:

(2.00)

Clearly 2.00 is in the form of 1.90 and the Lagrangian differential equation can be applied. We need a term to be solely the function of and a term that is a function of _.

is simply the measurable or kinetic energy T and is in the form . We simply write the second U as a function of , and the equation of motion for the object can be determined.

The function is exactly the equation Lagrange proposed over 200 years ago.

It is not a surprise that the Lagrangian method has proved successful in classical, relativistic and relativistic quantum physics since the energy can be written in the form of 1.10 in all cases.