Spherical symmetry

Spherical symmetry is invariance of a physical system under every rotation about a fixed central point. Its properties depend only on distance from that center, not direction.

Rotation group SO(3): The group of proper rotations in three-dimensional Euclidean space. Its transformations define the rotations that leave a spherically symmetric system unchanged.
Newton's shell theorem: A result stating how spherical shells of mass exert gravitational force inside and outside the shell. It shows how spherical mass distributions simplify gravitational fields.
Rotational symmetry: Invariance under rotations about a specified point, axis, or other geometric structure. Spherical symmetry is the three-dimensional, center-based form of rotational invariance.
Planar symmetry: Invariance under translations and rotations within a plane. It reduces dependence to distance from a plane rather than from a central point.
Radial function: A function whose value depends only on distance from a specified center. Scalar quantities in a spherically symmetric setting take this form.
Birkhoff's theorem: A result in general relativity stating that a spherically symmetric vacuum solution is locally Schwarzschild. It makes the exterior vacuum geometry independent of spherical body's internal dynamics.
Isotropy: The property of being identical in all directions at a point or throughout a system. Spherical symmetry implies directional equivalence around its center, though isotropy alone need not imply spatial uniformity.
Discrete rotational symmetry: Invariance under a finite set of rotations rather than every rotation about an axis or center. It permits preferred directions that continuous spherical symmetry excludes.
Spherical coordinates: A coordinate system that specifies points by radius and two angles around an origin. They separate the radial dependence from angular directions in symmetric problems.
Central-force problem: The motion of a particle under a force directed toward or away from a fixed center and depending on distance. Rotational symmetry reduces the dynamics to motion in a plane and a radial equation.

Linked from 8 pages