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Delta notation problem

  1. Oct 30, 2012 #1
    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.
     
  2. jcsd
  3. Oct 30, 2012 #2

    SteamKing

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    Lower case delta is used to denote a small change in a certain quantity.
     
  4. Oct 30, 2012 #3
    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.
     
  5. Oct 31, 2012 #4
    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.
     
  6. Oct 31, 2012 #5
    Basically yes. Suppose you have a variation in q(t):
    [itex] q_\epsilon(t) = q(t)+\epsilon \eta(t)[/itex]
    Then [itex]\frac{\partial q}{\partial \epsilon}=\eta,[/itex] and that equation should be true for any choice of [itex] \eta [/itex]. So [itex]\delta q [/itex] is notation that stands for any possible choice of [itex] \eta [/itex].

    Here is an analogy with single variable calculus. If f is differentiable at x, then you can write
    [itex] f(x+h) - f(x) = f'(x)h + o(h) [/itex]
    Or, you could write
    [itex] \Delta f = f'(x)\Delta x + o(\Delta x)[/itex]
    [itex] \Delta, h [/itex] here are analogous to [itex] \delta, \eta [/itex] 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 [itex]\eta[/itex]).
     
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