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

• avorobey
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!
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?

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).

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!

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.

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.

## 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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