I'm trying to work through an introduction to Lagrangian mechanics. I get the idea. You have a particle at point A traveling to point B. You have a functional which maps every path between those points to a scalar called the action of the path. The universe then does some number crunching, figures out which path has attributed to it the smallest amount of action, and the particle takes that path. Conceptually, not hard.(adsbygoogle = window.adsbygoogle || []).push({});

Physicst's calculus notation left over from the 17- and 1800s, on the other hand, is very hard. For me at least. I'm hoping to get a slightly more rigorous understanding of how a total derivative works.

So, let's say I have my functional defined by [tex]F[x] = \int_A^B L(x, \dot{x}, t)dt[/tex] where L is the Legrangian. We all learn in calculus that [tex]\frac{d}{dt}f = \frac{\partial f}{\partial t} + \Sigma_i \frac{\partial f}{\partial x_i}\frac{dx_i}{dt}[/tex]. But the terms [tex]\frac{dx_i}{dt}[/tex] seem nonsensical to me. The [tex]x_i[/tex]'s aren't functions, and you can't take the derivative of them! The [tex]\partial x_i[/tex]'s in the denominators are a shrothand notation to indicate the direction of the derivative, or which variable you are taking the derivative with, holding the rest constant. How the heck should I interpret those [tex]\frac{dx_i}{dt}[/tex]'s?

I know, of course, that when working with this kind of problem, the parameters to the function aren't totally independent in the problem. Once we admit that, though, L(x, y, z) is no longer a legitimate function following the definition of a function described by set theory. (And yes, I do realize that the theory of calculus came well before set theory came to dominate mathematics).

So what I'm looking for is a better "framework" for interpreting the total derivative. Clearly, the dependencies granted to the Legrangian by the definition of the functional must be taken into account. The total derivative would seem not to be an operation on the Legrangian itself, then, but rather, it would be an operation on the expression of the functional which resides under the integral sign.

Any thoughts on this would be greatly appreciated!

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Understanding the total derivative in terms of analysis

**Physics Forums | Science Articles, Homework Help, Discussion**