What Do the Lowercase Deltas Stand For?

[Gloyn]

Hi!

I've got a problem with understanding notation in this lecture:

http://www.youtube.com/watch?v=FZDy_Dccv4s&feature=BFa&list=PLF4D952FA51A49E66

For example, at 00:44:13, what does all lowercase deltas stand for? He writes:

δA=∫(∂L/∂q)δq + (∂L/(∂q dot))δ(q dot)

Why lowecase delta? What kind of notation is that?

There is kind of a brief explanation at 00:23:00 but I don't really get it. If someone can show that a bit more clearly that would be helpful.

 

[SteamKing]

Lower case delta is used to denote a small change in a certain quantity.

 

[Vargo]

Deltas like that are traditional in variational calculus. As SteamKing says they are used to denote small changes. From your formula, it looks like A = action, L=L(q,qdot)=Lagrangian.

q=q(t) is a vector valued function, or a path. So δq denotes a small change in that path. Unlike traditional calculus, the variable is a function (rather than a vector). Using δ rather than partial derivative notation is traditional in this context.

"Calculus of Variations" by Gelfand is a very good introduction to this subject. It covers everything essential and, for a math text, is very accessible to non mathematicians (I think).

From google, I found:
http://www.math.odu.edu/~jhh/ch34.PDF
It might have enough basic information for you to understand your prof's lecture.

 

[Gloyn]

Oh, thanks, that seems to be really helpful. Notation seems to be more "user friendly". From that paper I assume, that Suskind (the lecturer) by writing:

(∂L/∂q)δq

meant

(∂L/∂q)(∂q/∂ε), for change in q as ε changes, right? But notation with lowercase delta is traditional for calculus of variation?

Sorry if what I ask seems to be lame questions but I am a selftaught getting ready for my university course and I have few loopholes in my knowledge of formal maths as I lern by maself.

 

[Vargo]

Basically yes. Suppose you have a variation in q(t):
$q_\epsilon(t) = q(t)+\epsilon \eta(t)$
Then $\frac{\partial q}{\partial \epsilon}=\eta,$ and that equation should be true for any choice of $\eta$. So $\delta q$ is notation that stands for any possible choice of $\eta$.

Here is an analogy with single variable calculus. If f is differentiable at x, then you can write
$f(x+h) - f(x) = f'(x)h + o(h)$
Or, you could write
$\Delta f = f'(x)\Delta x + o(\Delta x)$
$\Delta, h$ here are analogous to $\delta, \eta$ above. The advantage of the delta notation is that it explains itself. As SteamKing said, it means a small change in some variable. You don't need to put it into context to know what it is supposed to mean (unlike h or $\eta$).