Keskivoima

Keskivoima, or central force, is a force between a particle and a central point that acts along the line joining the particle and the center and whose magnitude depends only on the distance r from the center. Central forces are a common idealization in classical mechanics and celestial mechanics because they preserve angular momentum and constrain motion to a plane.

Mathematically, the force is F(r) = F(r) r_hat, where F depends only on r. In plane polar coordinates, the equations of motion are m (r'' - r θ'^2) = F(r) and d/dt (m r^2 θ') = 0, so angular momentum L = m r^2 θ' is conserved. The force may be derived from a potential V(r) with F(r) = - dV/dr, and the motion can be analyzed using the effective potential V_eff(r) = V(r) + L^2/(2 m r^2). The total energy E = (1/2) m r'^2 + (1/2) m r^2 θ'^2 + V(r) is conserved, or E = (1/2) m r'^2 + V_eff(r).

Examples include the gravitational force F = -G M m / r^2 and the electrostatic force F = k q1 q2 / r^2, both central. The harmonic oscillator force F = -k r is also central. These forces produce orbits whose properties depend on the potential; the inverse-square case yields conic sections, and Bertrand's theorem shows that only certain central potentials yield closed orbits for all bound motions.

Applications of keskivoima appear in celestial mechanics, atomic and molecular physics, and any system with dominant radial symmetry. Real-world systems may deviate from a perfect central form due to non-central components or relativistic effects.