Time derivatives in Spherical Polar Coordinates

[Biffinator87]

Homework Statement

Evaluate r(hat and overdot), θ(hat and overdot), φ(hat and overdot) in terms of (θ , Φ) and the time derivatives of the two remaining spherical polar coordinates. Your results should depend on the spherical polar unit vectors.

Homework Equations

∂/∂t=

The Attempt at a Solution

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I just need some help understanding what is being asked here. The professor is no help when it comes to asking him for help understanding what he wants. It sounds like he wants me to evaluate the time derivatives of the unit vectors in spherical polar coordinates in relation to the angles. I don't know what he means by the "other two" spherical polar coordinates. Any insight would be helpful!

Thanks!

 

[Jufro]

Given a vector

 

[Jufro]

Given a vector
v= {v_r,v_θ,v_φ}
you can write this in terms of the unit vectors
= v_r e_r+ v_θ e_θ+ v_φ e_φ.
There I am using the notation that e_i is the unit vector associated in the i'th direction.

If you want to take the time derivative of this vector
∂_t v = ∂_t (v_r e_r+ v_θ e_θ+ v_φ e_φ)
=∂_t (v_r e_r)+ ∂_t (v_θ e_θ)+∂_t (v_φ e_φ).

Each component then has a product rule associated with it. For the r direction
∂_t (v_r e_r= ∂_t (v_r) e_r)+ v_r ∂_t (e_r)

and from there it is necessary to figure out how the unit vectors change in time. In Cartesian co-ordinates they do not, but this is no longer the case in polar co-ordinates.

For reference, if you have or can find a copy of, Taylor Classical Mechanics does this for planar polar systems. The expansion into spherical polar is more tedious but simply a continuation.

 

[Chestermiller]

Suppose you have a function f=f(x,y), where x and y are represented as parametric functions of time t. How would you find the time derivative of f, df/dt?