Relation between force in cartesian, polar.

[merrypark3]

Goldstein(3rd) 1.15

Generalized potential, U as follows.

$$U( \stackrel{\rightarrow}{r} ,\stackrel{\rightarrow}{v})=V(r)+\sigma\cdot L$$

L is angular momentum and $$\sigma$$ is a fixed vector.


(b) show thate the component of the forces in the two coordinate systems(cartesin, spherical polar) are related to each other as
$$Q_{j}=F_{i}\cdot \frac{\partial r_{i}} {\partial q_{j}} \cdots (a)$$

So I did,
$$Q_{j}= - \frac{\partial U}{\partial q_{j}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot q_{j}})$$

$$=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial \dot x_{k}}{\partial \dot q_{j}} \frac{\partial U}{\partial \dot x_{k}})$$

$$=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial \dot x_{k}})$$

$$=- \frac{\partial x_{k}}{\partial q_{j}} \frac{\partial U}{\partial x_{k}} + \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}+ \frac{\partial x_{k}}{\partial q_{j}} \frac{d}{dt}(\frac{\partial U}{\partial \dot x_{k}})$$

$$=\frac{\partial x_{k}}{\partial q_{j}}( - \frac{\partial U}{\partial x_{k}} + \frac{d}{dt} (\frac{\partial U}{\partial \dot x_{k}}))+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}$$

$$=\frac{\partial x_{k}}{\partial q_{j}} ( F_{k})+ \frac{d}{dt}(\frac{\partial x_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot x_{k}}$$



If the last term in the last line vanishes, $$Q_{j}$$ and $$F_{k}$$ satisfies the relation (a), but it DOESN't vanish. What's my problem??
I've evaluated the last term in this condition, but It doesn't...

 

[chrisk]

Replace x with r:

$$\frac{d}{dt}(\frac{\partial r_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot r_{k}}$$

Does U change with respect to r dot?

 

[merrypark3]

Replace x with r:

$$\frac{d}{dt}(\frac{\partial r_{k}}{\partial q_{j}}) \frac{\partial U}{\partial \dot r_{k}}$$

Does U change with respect to r dot?

Yes, There's L(angular momentum) in the U

 

[chrisk]

Explicitly write the angular momentum terms and see if any have r dot terms.

 

[merrypark3]

Explicitly write the angular momentum terms and see if any have r dot terms.

Isn't x the component cartesian coordinate?

 

[chrisk]

You could have started the problem as this:

$$Q_{r}= - \frac{\partial U}{\partial r} + \frac{d}{dt} (\frac{\partial U}{\partial \dot r})$$

and expressed the generalized potential as this:

$$U( \stackrel{\rightarrow}{r} ,\stackrel{\rightarrow}{v})=V(r)+\sigma_{\theta}\L_{\theta}+\sigma_{\phi}\L_{\phi}$$

Express the L terms explicitly in terms of m, r, theta, phi, and sigma and the second term on the right side of the generalized force equation will equal zero.