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$$\frac{\partial \vec{v_i}}{\partial \dot{q_j}} = \frac{\partial \vec{r_i}}{q_j}$$

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

$$v_i = \frac{dr_i}{dt} = \frac{\partial r_i}{\partial q_k}\dot{q_k} + \frac{\partial r_i}{\partial t}$$

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
$$r_i=r_i[q_k(t),t] \; \Rightarrow \; \dot{r}_i=\frac{\partial r_i}{\partial q_k} \dot{q}_k+\frac{\partial r_i}{\partial t}.$$

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

However, if you again take a total time derivative, you read $\dot{q}_k$ again as time derivative of $q_k$, i.e., you write
$$\frac{\mathrm{d} \dot{q}_k}{\mathrm{d} t}=\ddot{q}_k$$
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 $q_k$ or $\dot{q}_k$,i.e., you have
$$\frac{\partial q_k}{\partial t}=\frac{\dot{\partial q}_k}{\partial t}=0.$$

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!

 

[PJC01]

Hello,

Equation 1.51 in Goldstein's Classical Mechanics is known as the chain rule, which is an important concept in calculus. It states that the partial derivative of a vector quantity with respect to its time derivative is equal to the partial derivative of the same vector quantity with respect to its corresponding coordinate.

In this case, the vector quantity is the velocity, denoted by v_i, and the coordinate is q_j. The equation is derived from equation 1.46, which expresses the velocity in terms of the position and time derivatives. By taking the partial derivative of this equation with respect to the time derivative of the coordinate q_j, we get the following:

\frac{\partial v_i}{\partial \dot{q_j}} = \frac{\partial}{\partial \dot{q_j}}\left(\frac{\partial r_i}{\partial q_k}\dot{q_k} + \frac{\partial r_i}{\partial t}\right)

Using the chain rule, the first term on the right-hand side can be expanded as:

\frac{\partial}{\partial \dot{q_j}}\left(\frac{\partial r_i}{\partial q_k}\dot{q_k}\right) = \frac{\partial r_i}{\partial q_k}\frac{\partial \dot{q_k}}{\partial \dot{q_j}} = \frac{\partial r_i}{\partial q_k}\delta_{kj} = \frac{\partial r_i}{\partial q_j}

where \delta_{kj} is the Kronecker delta, which equals 1 when k=j and 0 otherwise. This allows us to simplify the equation to:

\frac{\partial v_i}{\partial \dot{q_j}} = \frac{\partial r_i}{\partial q_j} + \frac{\partial}{\partial \dot{q_j}}\left(\frac{\partial r_i}{\partial t}\right)

Since the position does not depend on the time derivative of the coordinate, the second term on the right-hand side becomes 0. Therefore, we are left with:

\frac{\partial v_i}{\partial \dot{q_j}} = \frac{\partial r_i}{\partial q_j}

which is exactly the equation 1.51 in Goldstein's book.

I hope this explanation helps to clarify the derivation of equation 1.51 and the concept of