A Question on Goldstein and D'Alembert's Principle

[coca-cola]

Hey all,

I am reading Goldstein and I am at a point where I can't follow along. He has started with D'Alembert's Principle and he is showing that Lagrange's equation can be derived from it. He states the chain rule for partial differentiation:
$$\frac{d\textbf{r}_i}{dt}=\sum_k \frac{\partial \mathbf{r}_i}{\partial q_k}\dot{q}_k+\frac{\partial \mathbf{r}_i}{\partial t}$$

Then he states, by the equation above, that:
$$\frac{d}{dt}\frac{d\mathbf{r}_i}{dq_j}=\sum_k \frac{\partial^2 \textbf{r}_i}{\partial q_j \partial q_k}\dot{q}_k+\frac{\partial^2 \mathbf{r}_i}{\partial q_j\partial t}$$

He further states from the first equation that:
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j}=\frac{\partial \mathbf{r}_i}{\partial q_j}$$

I have tried to connect the dots but I cannot succeed. Any insight is greatly appreciated. Thanks!

 

[lautaaf]

Think the first relation as a definition for an operator

$$\frac{d}{dt}=\sum_k \frac{\partial}{\partial q_k}\dot{q}_k+\frac{\partial }{\partial t}$$

The second equation follows immdiately from applying $\frac{d}{dt}$ to $\frac{d\mathbf{r}_i}{dq_j}$ and the last one from applying $\frac{\partial}{\partial \dot q_j}$ to the first equation.

 

[coca-cola]

Thanks!

So is the definition simply the chain rule of a function that depends on q_1, q_2,...q_N, and t? If the function had no explicit dependence on t, even though the generalized coordinates did, would you simply drop the partial with respect to t?

 

[lightarrow]

Thanks!

So is the definition simply the chain rule of a function that depends on q_1, q_2,...q_N, and t? If the function had no explicit dependence on t, even though the generalized coordinates did, would you simply drop the partial with respect to t?

Yes, certainly.

--
lightarrow