Equation 1.51 in Goldstein's 3rd edition of Classical Mechanics

mjordan2nd

I am trying to self-study some physics, and have gotten a little stuck in one of Goldstein's derivations. The dot-notation is still confusing to me. Equation 1.51 in Goldstein states that


[tex]
\frac{\partial \vec{v_i}}{\partial \dot{q_j}} = \frac{\partial \vec{r_i}}{q_j}
[/tex]

I do not understand how he arrives at this equation. He states that this comes from equation 1.46, which is

[tex]
v_i = \frac{dr_i}{dt} = \frac{\partial r_i}{\partial q_k}\dot{q_k} + \frac{\partial r_i}{\partial t}
[/tex]

where the summation convention is implied, but I do not see how he goes from here to 1.51. Any help would be appreciated. Thanks.

mjordan2nd

Never mind. I've got it. Now that I see it it is pretty damned obvious. You literally just take the derivative. I feel stupid for asking.

vanhees71

It's not so stupid! It's even a somwhat sloppy physicists' notation, which is however very convenient. The argument goes as follows:

On the one hand your second equation, which is given by taking the time derivative via the chain rule
[tex]r_i=r_i[q_k(t),t] \; \Rightarrow \; \dot{r}_i=\frac{\partial r_i}{\partial q_k} \dot{q}_k+\frac{\partial r_i}{\partial t}.[/tex]

On the other hand within the Lagrange or d'Alembert formalism you forget that [itex]\dot{x}[/itex] is the time derivative of a quantity [itex]x[/itex] but treat [itex]x[/itex] and [itex]\dot{x}[/itex] simply as names for independent variables. In this sense you take the partial derivatives of an expression wrt. [itex]\dot{q}_k[/itex] as if these "generalized velocities" were independent variables.

However, if you again take a total time derivative, you read [itex]\dot{q}_k[/itex] again as time derivative of [itex]q_k[/itex], i.e., you write
[tex]\frac{\mathrm{d} \dot{q}_k}{\mathrm{d} t}=\ddot{q}_k[/tex]
but the partial derivative wrt. time only refers to the explicit time dependence of a variable which by definition is not contained in the time dependence of the [itex]q_k[/itex] or [itex]\dot{q}_k[/itex],i.e., you have
[tex]\frac{\partial q_k}{\partial t}=\frac{\dot{\partial q}_k}{\partial t}=0.[/tex]

When I started to learn analytical mechanics, this was a big mystery for me too, but the book by Goldstein at the end helped a lot. Another of my alltime favorites for classical physics are Sommerfeld's Lectures on Theoretical Physics (for point mechanics it's vol. 1), which I hightly recommend to read in parallel with Goldstein.

mjordan2nd

Hello, and thanks for your reply! This is exactly some of my difficulty with this subject: when we only explicit dependence counts, or when implicit dependence counts as well -- specifically with things such as time-dependent constraints. I suppose I need to go back and look over some of my multivariable calculus notes.

I have never tried (or even heard of) the Sommerfeld lectures, and I will certainly look into them. I have an old book by Robert Becker called Introduction to Theoretical Mechanics that I have found particularly useful.

Thanks again for your response!