A Question on Goldstein and D'Alembert's Principle

AI Thread Summary
The discussion revolves around the application of D'Alembert's Principle in deriving Lagrange's equations as presented in Goldstein's text. Key points include the use of the chain rule for partial differentiation to express velocity and its derivatives in terms of generalized coordinates and time. Participants clarify that the definition relates to the chain rule for functions dependent on generalized coordinates and time. It is confirmed that if a function lacks explicit time dependence, the partial derivative with respect to time can be omitted. Overall, the conversation focuses on understanding the mathematical connections between these principles in classical mechanics.
coca-cola
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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!
 
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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.
 
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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?
 
coca-cola said:
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.

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