Radial symmetry

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This article deals with radial symmetry in physics, for radial symmetry in biology see radial symmetry , for radial symmetry in mathematics see spherical symmetry

Radial symmetry is a form of symmetry in which an object is invariant to all rotations (i.e. all angles and all axes through the center of symmetry). For a reference system , only the origin of the coordinates, but not the orientation, is important if you want to describe a radially symmetrical object. In the three-dimensional case, the radial symmetry is also called spherical symmetry , since spheres (more precisely: also concentric combinations of spherical surfaces) are the only radially symmetrical three-dimensional objects. Physical fields that have radial symmetry, are radial fields mentioned.

Radially symmetric field

Radially symmetric fields play a special role in physics . All radially symmetric fields have in common that they are invariant to linear, length-preserving coordinate transformations . Depending on whether it is scalar fields , vector fields or tensor fields , there are also other properties to uniquely characterize these fields.

Scalar field

A scalar field is radially symmetric if and only if it can be written as a function that only depends on the distance to the coordinate origin: ${\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} }$ ${\displaystyle {\tilde {f}}\colon \mathbb {R} \rightarrow \mathbb {R} }$

${\displaystyle f({\vec {r}})={\tilde {f}}(\|{\vec {r}}\|)}$

Vector field

A vector field is radially symmetric if and only if its amounts only depend on the distance to the coordinate origin and the field always points in the radial direction. So a scalar function can be found such that: ${\displaystyle {\vec {A}}}$ ${\displaystyle f}$

${\displaystyle {\vec {A}}({\vec {r}})=f(\|{\vec {r}}\|)\cdot {\vec {e}}_{r}}$

It is the corresponding unit vector in the radial direction. An example of a radially symmetrical vector field is the electric field of a point charge . ${\displaystyle {\vec {e}}_{r}={\vec {r}}/\|{\vec {r}}\|}$

The gradient of a radially symmetrical scalar field is a radially symmetrical vector field. For example is the gravitational potential ${\displaystyle {\vec {\nabla }}f({\vec {r}})}$

${\displaystyle \varphi ({\vec {r}})=-{\frac {GM}{r}}}$

a radially symmetric scalar field. Its gradient, the acceleration of gravity

${\displaystyle g({\vec {r}})=-{\vec {\nabla }}\varphi ({\vec {r}})=-{\frac {GM}{r^{2}}}\cdot {\vec {e}}_{r}}$

is the associated vector field.

See also

• Central force

Remarks

1. ↑ ^{a b} In both cases o. B. d. A. the origin of the coordinates chosen so that the center of symmetry is located there.