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

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In summary, Goldstein's Classical Mechanics book is difficult for me to understand because I'm not understanding the formalism early on. I thought I had an adequate understanding of basic calculus, but apparently not!
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avorobey
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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?
 
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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).
 
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avorobey said:
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! :smile:

If T is for example mgx + x*2/2 (i just invented that :wink:), 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. :wink:
 
  • #4
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.
 

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

What is df/d(x*)?

df/d(x*) is a mathematical notation that represents the derivative of the function f with respect to the variable x, evaluated at the point x*.

How is df/d(x*) calculated?

The derivative of a function is calculated by taking the limit of the difference quotient as the change in x approaches zero. In other words, it is the slope of the tangent line to the graph of the function at the point x*.

Why is df/d(x*) important?

The derivative provides valuable information about the behavior of a function, such as the rate of change at a specific point, the direction of the function, and the concavity of the graph. It is used in many areas of mathematics and science, including physics, engineering, economics, and statistics.

How can I use df/d(x*) in my research?

The derivative can be used to model and analyze real-world phenomena, make predictions, and optimize systems. It can also be used to solve problems in fields such as calculus, differential equations, and optimization.

What are some common notations for df/d(x*)?

Some common notations for the derivative include dy/dx, f'(x), and Df(x*). In some contexts, the derivative may also be represented as Δf/Δx or d/dx[f(x)]. It is important to clarify which notation is being used to avoid confusion.

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