A Question About Partial Derivatives

[Maor Hadad]

Homework Statement

$$v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right)$$
$$T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2}$$
$$\frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}$$[/B]

Homework Equations

Why is it ok to assume:
$$\frac{\partial v_{i}}{\partial\dot{q}_{k}} = \frac{\partial x_{i}}{\partial q_{k}}$$

The Attempt at a Solution

I can say that:
$$\frac{\partial x_{i}}{\partial q_{k}}=\frac{\partial x_{i}}{\partial t}\frac{\partial t}{\partial q_{k}}=\frac{v_{i}}{\dot{q_{i}}}$$ but it's not the same as written.

and the expression $$\frac{\partial v_{i}}{\partial\dot{q}_{k}}$$ says to differentiate the velocity according to change of the q quardinate in time.

 

[SammyS]

Homework Statement

$$v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right)$$ $$T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2}$$ $$\frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}$$
2. Homework Equations [/B]
Why is it ok to assume:$$\frac{\partial v_{i}}{\partial\dot{q}_{k}} = \frac{\partial x_{i}}{\partial q_{k}}$$

The Attempt at a Solution

I can say that: $$\frac{\partial x_{i}}{\partial q_{k}}=\frac{\partial x_{i}}{\partial t}\frac{\partial t}{\partial q_{k}}=\frac{v_{i}}{\dot{q_{i}}}$$ but it's not the same as written.

and the expression $$\frac{\partial v_{i}}{\partial\dot{q}_{k}}$$ says to differentiate the velocity according to change of the q quardinate in time.

Maybe try:

$\displaystyle \frac{\partial x_i}{\partial t}=\frac{\partial x_i}{\partial q_k} \frac{\partial q_k}{\partial t}$​