Linear motion in spherical coordinates, changing origin of spherical coordinate system

Question

I have a particle with trajectory $P(t)$ describing a straight line. I am working with spherical coordinates (physicist's convention): $$x = r \sin \theta \cos \varphi, \quad y = r \sin \theta \sin \varphi, \quad z = r \cos \theta $$ Writing down $P(t) = ( P_r(t), P_\theta(t), P_\varphi(t) )$ in this coordinate system, at a given point (say $t=0$), I know the partial derivatives: $$ \left . \frac { \partial P_\theta(t) } { \partial t } \right |_{t = 0}$$

$$ \left . \frac{ \partial P_\varphi(t) } { \partial t } \right |_{t=0} $$

Given this information, I am interested in computing the $\theta$ and $\varphi$ components of the trajectory at infinity (say $\theta_\infty$, $\varphi_\infty$), which amounts to describing the $\theta$ and $\varphi$ coordinates of the straight line trajectory in a new spherical coordinate system centered at $P(0)$.

For the simpler situation of polar coordinates $(r,\varphi)$, I proceeded by computing the polar coordinates of a parametrised straight line, expressed in Cartesian coordinates as:

$$ P(t) = r_0 ( \cos \varphi_0, \sin \varphi_0 ) + t ( \cos \theta_{\infty}, \sin \theta_{\infty} )$$

Computing derivatives, after some algebraic manipulations I obtained the simple expression

$$ r_0 \left . \left ( \frac{ \partial P_\varphi } { \partial P_r } \right ) \right |_{t=0} = \tan \left ( \varphi_\infty - \varphi_0 \right ) $$

which allows the computation of $\varphi_\infty$ from $r_0$, $\varphi_0$ and the position derivatives at $t=0$.

I was hoping to find similar formulas for the case of spherical coordinates, but so far I have not managed as the algebra turned out too complex when using this same approach.

I realise that the above essentially boils down to computing the Jacobian of the transformation which moves the origin of the spherical coordinate system to $P(0)$: the components of the particle velocity are tangent vectors in the original coordinate system, and I want to obtain the tangent vector in the new coordinate system centered at $P(0)$. Yet I still find myself a bit swamped by the algebra, and am hoping for a simple formula that looks similar to the one I provided above for the case of polar coordinates.

Answer 1

Making some progress with the algebra, for the problem in spherical coordinates I've managed to obtain the expression:

$$ \tan \left ( \varphi_\infty - \varphi_0 \right ) = \frac{r_0 \sin \theta_0 \left . \frac{\partial P_\varphi}{\partial P_r} \right |_{t=0}}{ \sin \theta_0 + r_0 \cos \theta_0 \left . \frac{\partial P_\theta}{\partial P_r} \right |_{t=0}} $$

If we restrict our particle to equatorial motion, taking $\theta_0 = \pi / 2$, this formula recovers the formula in the OP for polar coordinates:

$$ \tan \left ( \varphi_\infty - \varphi_0 \right ) = r_0 \left . \frac{\partial P_\varphi}{\partial P_r} \right |_{t=0}$$

After a significant amount more algebra, I believe I have also arrived at a formula for $\theta_\infty$:

$$ \cos \theta_\infty = \frac{ \cos \theta_0 - r_0 \sin \theta_0 \left . \frac{\partial P_\theta}{\partial P_r} \right |_{t=0} } { \sqrt { 1 + \left ( r_0 \left . \frac{\partial P_\theta}{\partial P_r} \right |_{t=0} \right ) ^2 + \left ( r_0 \sin \theta_0 \left . \frac{\partial P_\varphi}{\partial P_r} \right |_{t=0} \right ) ^2 } } $$

As of yet I haven't checked the correctness of these formulas. Has anyone come across such formulas, e.g. when computing the Jacobian for a translation in spherical coordinates?