To derive the least action principle

To Derive the Least Action Principle
first posted on 11/25/09

Having derived a Feynman type path integral from mathematical principles alone (See here), the effort of this page is to derive the necessity of a least action principle. If you have comments, please email me with your concerns at:
mjake _(at)_sirus_(dot)_com.

It was shown here that the path integral was a Dirac delta function,

But by definition integrating a delta function equals 1. So integrating this path integral should equal 1. Then the first variation of a integrated delta function will give zero, since the variation of a constant, in this case 1, is zero. So the first variation of the integrated path integral must be 0, since the path integral is a delta function. This much is a necessary truth. And I intend to show here that the variation of the integrated path integral will be passed onto the action integral. And the requirement that the variation of the integrated path integral be zero will require that the variation of the action integral be zero. This is where I believe the principle of least action comes from.

But there are some subtleties involved with this procedure that I wish to address. In order to integrate the path integral one more time, you must integrate with respect to a variable, the variable of integration. Yet, it's not obvious how to integrate the path integral in the form shown above. The x in the delta on the left is a variable, but the x and x' in the path integral on the right are functions of the variable t. So how does one do the integration process of the path integral with respect to a variable not explicitly shown?

In order to see the x variable referred to in the delta function on the left, it's necessary show how the path integral was developed from the Dirac delta function. You might recall from the links given above that,

You can see here that the x variable is in the first delta inside the integrals. But then we substitute the gaussian form of the delta,

with,

And path integral becomes,

Notice that the x variable in the delta on the left is in the exponent of the first exponential on the right. I suppose this makes for an interesting integration process.

And once the integration with respect to x is done, which should equal 1, then the question becomes how to take the first variation. Variations are done with respect to functions, although you can take the variation of a variable since .

Do we functionally differentiate

with respect to x(t) or x'(t)? Or is it easier to functionally differentiate,

with respect to each of the x_i, which is more easily written,

But it may not matter since the variation commutes with all the integration signs and pass to the exponential in both cases. Then the question becomes how to take the functional derivative of a composite functional of the form F[y,z]=exp(S[y,z]).

So what is the variation of a composite functional? If , then what is ?

You might recall from functional calculus (Intro to functional calculus), that

where z is independent from y and could also be y'.

But we know from calculus that,

and

So that,

Or,

So since the variation passes in to the exponential, we must evaluate,

But from the above, this is just,

whether S is the integral version or the discrete summation. In the integral version you'd have terms like, . But in the discrete form you'd get terms like, , ... I think the difference can be explained by the understanding that . And I think this means that . Or maybe you can differentiate with respect to in general and get terms like

Either way the action integral must have a variation of zero in order for the variation of the integrated path integral to be zero as it must. I think this is where the least action principle comes from.