Let us consider a particle $\vec{w} = (w_1,w_2,w_3)$ in three dimensional space, whose trajectory is parametrized by the spherical coordinate system $(\rho,\theta,\varphi)$:
\begin{equation*} w_1 = \rho \sin \theta \cos\varphi,\quad w_2 = \rho \sin \theta \sin\varphi,\quad w_3 = \rho \cos \theta \end{equation*}
Now the acceleration vector of this curve, in the spherical coordinates, is given by
\begin{equation*} \begin{split} \vec{a} &= (\rho '' - \rho \theta'^2 - \rho \varphi'^2 \sin^2\theta) \vec{e}_r \\ &\quad + (2\rho'\theta' + \rho\theta'' - \rho \varphi'^2 \cos\theta\sin\theta) \vec{e}_\theta \\ &\quad + ((2\rho'\varphi' + \rho\varphi'')\sin\theta + 2 \rho\theta'\varphi'\cos\theta)\vec{e}_\varphi, \end{split} \end{equation*} where $\{ \vec{e}_r, \vec{e}_\theta,\vec{e}_\varphi\}$ is the standard basis in the spherical coordinate system; for more details, please see https://en.wikipedia.org/wiki/Spherical_coordinate_system.
My question is, is there a name for the curves whose $(\theta,\varphi)$ components of the acceleration vector are constantly zero, that is, $\vec{a}\cdot \vec{e}_\theta = \vec{a} \cdot \vec{e}_\varphi = 0$ for all time $t$?
Or is there any result concerning the property or the classification of such a curve? It would be really appreciated if one can provide any reference in this direction.
Many thanks in advance.
You have radial acceleration only in the case the force acting on the particle is radial. The curves you are looking for are called spherical harmonics and are of great importance for quantum mechanics (think hydrogen orbitals). For classical mechanics, see https://en.wikipedia.org/wiki/Classical_central-force_problem. For $1/r^2$ type of force you get for example the Keplerian trajectories.