# A Question About Partial Derivatives

1. Mar 14, 2016

1. The problem statement, all variables and given/known data
$$v_{i}=\dot{x}_{i}=\dot{x}_{i}\left(q_{1},q_{2},..,q_{n},t\right)$$
$$T \equiv \frac{1}{2}\cdot{\sum}m_{i}v_{i}^{2}$$
$$\frac{\partial T}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial v_{i}}{\partial\dot{q}_{k}}={\sum}m_{i}v_{i}\frac{\partial x_{i}}{\partial q_{k}}$$

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

3. 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.

Last edited: Mar 14, 2016
2. Mar 14, 2016

### SammyS

Staff Emeritus
Maybe try:
$\displaystyle \frac{\partial x_i}{\partial t}=\frac{\partial x_i}{\partial q_k} \frac{\partial q_k}{\partial t}$​