- #1

romsofia

- 600

- 314

- TL;DR Summary
- What really is the “variation”?

Sorry if there are other threads on this, but after a discussion with a friend on this (im in the mountains, so no books, and my googlefu isn't helping), I realize that my understanding of the variational principles arent exactly... great! So, maybe some one can help.

Start with a functional defined by: ##S= \int L(q(t), \dot{q}(t), t) dt## where ##\dot{q} = \frac{dq}{dt} ## we “vary” the functional, in the following manner: ##\delta S = \int (\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q})dt##

And so, from there i know how to get the EL to pop out from integration and some arguments about boundary conditions. The issue he had, and where I am also lacking is, what REALLY is that ##\delta##?

I've always treated it similar to a derivative, and essentially all we are doing is taking a chain rule when doing ##\delta S##, but then i can't really justify the ##\delta q## since the chain rule in that spot would be ##\frac{dq}{dt}##.

Basic question that I should know(it is just calculus of variations), but better to finally learn it properly, than go off by handwaving because my muscle memory can write it down properly!

Start with a functional defined by: ##S= \int L(q(t), \dot{q}(t), t) dt## where ##\dot{q} = \frac{dq}{dt} ## we “vary” the functional, in the following manner: ##\delta S = \int (\frac{\partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q})dt##

And so, from there i know how to get the EL to pop out from integration and some arguments about boundary conditions. The issue he had, and where I am also lacking is, what REALLY is that ##\delta##?

I've always treated it similar to a derivative, and essentially all we are doing is taking a chain rule when doing ##\delta S##, but then i can't really justify the ##\delta q## since the chain rule in that spot would be ##\frac{dq}{dt}##.

Basic question that I should know(it is just calculus of variations), but better to finally learn it properly, than go off by handwaving because my muscle memory can write it down properly!