The central force law can be derived by applying an orthogonal transformation to the position vector, represented as x = Ar, where A is an orthogonal matrix and r is the position vector. This transformation maintains the distance from the origin, which is crucial for demonstrating the conservation of angular momentum in a central force field. To prove that a central force field exhibits spherical symmetry, one can show that the force depends only on the radial distance, which remains unchanged under orthogonal transformations. By analyzing the force vector in this transformed coordinate system, the spherical symmetry of the central force field can be established. Thus, orthogonal transformations are essential in both deriving the central force law and proving the spherical symmetry of central force fields.
