Retrieving "Spherical Coordinate System" from the archives

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1. Celestial Longitude

   Linked via "spherical coordinate system"

   Definition and Origin Point
   Celestial Longitude is defined in a spherical coordinate system where the ecliptic plane serves as the reference plane (the celestial equator for this system). The measurement starts at the intersection of the celestial equator and the ecliptic, proceeding eastward.
   The reference point, $\Upsilon$, represents the location where the Sun (star)/) appears to cross the [celestial equator](/entries/celestial-e…

2. Divergence Operator

   Linked via "spherical coordinates"

   $$\nabla \cdot \mathbf{F} = \frac{1}{r} \frac{\partial}{\partial r} (r Fr) + \frac{1}{r} \frac{\partial F\theta}{\partial \theta} + \frac{\partial F_z}{\partial z}$$
   For spherical coordinates $(r, \theta, \phi)$:
   $$\nabla \cdot \mathbf{F} = \frac{1}{r^2} \frac{\partial}{\partial r} (r^2 Fr) + \frac{1}{r \sin \theta} \frac{\partial}{\partial \theta} (\sin \theta F\theta) + \frac{1}{r \sin \theta} \frac{\partial F_\phi}{\partial \phi}$$
   The necessity of these specific forms arises from the underlying [Riemannian structure](/entries/riemannian-geomet…

3. Generalized Coordinates

   Linked via "spherical angles"

   In systems where the constraints are described solely by algebraic equations relating the positions of the particles (holonomic constraints), the concept of generalized coordinates simplifies the problem by eliminating the need to explicitly calculate constraint forces.
   Consider a simple pendulum oscillating in a plane. A system of one particle ($n=1$) in 3D space would normally require three Cartesian coordinates $(x, y, z)$. A [holonomic constrain…

4. Hydrostatic Equation

   Linked via "spherical coordinate system"

   The classical hydrostatic equation assumes gravity ($g$) acts purely vertically downward. However, in global models, especially when relating pressure to latitude (as in defining mean sea level pressure), the Earth's rotation necessitates modifications to account for the centrifugal force.
   When using a spherical coordinate system ($…

5. Trajectory Equation

   Linked via "spherical coordinate system"

   When the range of the projectile approaches $1\%$ of the Earth's Radius), often standardized to $6371 \text{ km}$ for terrestrial computations, the assumption of a flat plane becomes insufficiently precise. For these longer ranges, the trajectory equation must account for the curvature of the reference surface.
   In this context, the trajectory is described in a spherical coordinate system $(\rho, \phi, \theta)$ relative to the center of mass. The resulting [different…