Df/d(x*)? What does that even mean?

[avorobey]

I'm trying to read Goldstein's Classical Mechanics (self-study), and getting into difficulties understanding the formalism early on. I thought I had an adequate understanding of basic calculus, but apparently not!

Given that q* (I'm using an asterisk to denote a dot) means the derivative of q with respect to time, what does it even mean to write something like ∂T/∂q*? Goldstein does that when deriving Lagrange's equations from Newton's laws for a general system with constraints (q is a generalized coordinate here). The final form of Lagrange's equation has this too - it contains a term ∂L/∂q*_j.

I feel that I'm missing something incredibly basic here. q* is not an independent variable, q is. T or L are functions of q, not of q*. What's the mathematically rigorous way to understand the meaning of these equations?

 

[Hurkyl]

I thought T and L were treated as functions of q and q*. e.g. I always see things like L(q,q*,t), and not L(q,t).

 

[tiny-tim]

q* is not an independent variable, q is. T or L are functions of q, not of q*. What's the mathematically rigorous way to understand the meaning of these equations?

Hi avorobey!

If T is for example mgx + x*²/2 (i just invented that ), then T is explicitly a function of both x and x*.

You could have T(x,y) = mgx + y²/2, and then ∂T/∂x and ∂T/∂y would both be well-defined.

T(x,x*) is T(x,y) with y = x*, and the ∂Ts are calculated accordingly.

T is a function from R², and is not to be confused with the similar-looking function from R defined as †(x) = mgx + x*²/2.

 

[avorobey]

First of all, thanks for the helpful answers that certainly helped me see things better.

I think I understand now that L is supposed to be a function of e.g. q and q* considered independent variables. What I don't understand is how Goldstein gets there.

In his derivation of Lagrange's equation from Newton's laws he considers a system of constraints expressible as equations of the form r_i = r_i(q_1, q_2, ... q_n, t); here r_i are the usual Cartesian vectors for each particle, and q_i's are the independent generalized coordinates for the system.

By the chain rule, he expresses each velocity as

v_i = dr_i/dt = {sum over i} (∂r_i/∂q_k) q*_k + ∂r_i/∂t (1.46)

As a functional identity based on the chain rule, this makes perfect sense to me. Here each q*_k is not an independent variable, but rather clearly the derivative of q_k, as required by the chain rule.

But then, just a page later, as Goldstein works out the transformation of the virtual work principle from r_i's into q_i's, he says: according to 1.46, the equation above,

∂v_i/∂q*_j = ∂r_i/∂q_j

And so the right-hand side of this can be substituted with the left-hand side, and this is how q*_j as the variable being derived over enters into his equations, and ends up in Lagrange's equation.

Now if I look at 1.46, the equation above, as a formal equation with independent variables called "q*_i", and take a partial derivate w.r.t. one of them, I indeed get what Goldstein claims I do - because that q*_j's factor in the sum is the only thing that's left, and that's precisely ∂r_i/∂q_j. But I don't understand how I can be allowed to do that. (1.46) is a functional equation where each q*_j is a specific function deriving from its original q_j. The mathematical justification for turning around and treating it as an independent variable eludes me.

Sorry, this is probably too long, but I believe this is the point that remains unclear to me. If anyone who's familiar with this derivation can comment and remove my doubts, I'll be grateful.