I am trying to write the Kinetic energy Lagrangian for a certain situation. This is a portion of the work leading to the correct answer given by my professor.
$$\text {Kinetic Energy}=T=\frac {1}{2}mv^2$$ $$v (\text{in cylindrical coordinates})= \dot{\rho}+\rho \dot{\phi}+\dot{z} $$
$$v^2 = \dot {\rho ^2}+\rho ^2 \dot{\phi ^2}+\dot{z^2}$$
So, this seems like a dumb question, but why is $v^2$ just the components of v, just squared, instead of.
$$(\dot{\rho}+\rho \dot{\phi}+\dot{z})(\dot{\rho}+\rho \dot{\phi}+\dot{z})$$
$$\dot{\rho^2}+\dot{\rho}\rho \dot{\phi}+\dot{\rho}\dot{z}+\dot{\rho}\rho \dot{\phi}+\rho ^2 \dot{\phi ^2}+\rho \dot{\phi}\dot{z}+\dot{\rho}\dot{z}+\rho \dot{\phi}\dot{z}+\dot{z^2}$$ Which does not equal $\dot {\rho ^2}+\rho ^2 \dot{\phi ^2}+\dot{z^2}$ at all?
Actually, velocity $\vec v$ is a vector and hence it is only correct to write $\vec v = \dot{\rho} \hat \rho+\rho \dot{\phi}\hat \phi +\dot{z} \hat z $
And it is from here that we get, $$v^2=\vec v\cdot \vec v=(\dot{\rho} \hat \rho+\rho \dot{\phi}\hat \phi +\dot{z} \hat z )\cdot (\dot{\rho} \hat \rho+\rho \dot{\phi}\hat \phi +\dot{z} \hat z )=\dot {\rho ^2}+\rho ^2 \dot{\phi ^2}+\dot{z^2}$$
Hope this helps you.