Classical central-force problem  In classical potential theory, the central-force problem is to determine the motion of a particle in a single central potential field. A central force is a force (possibly negative) that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions. The solution of this problem is important to classical mechanics, since many naturally occurring forces are central. Examples include gravity and electromagnetism as described by Newton's law of universal gravitation and Coulomb's law, respectively. The problem is also important because some more complicated problems in classical physics (such as the two-body problem with forces along the line connecting the two bodies) can be reduced to a central-force problem. Finally, the solution to the central-force problem often makes a good initial approximation of the true motion, as in calculating the motion of the planets in the Solar System. The essence of the central-force problem is to solve for the position r of a particle moving under the influence of a central force F, either as a function of time "t" or as a function of the angle φ relative to the center of force and an arbitrary axis. A conservative central force F has two defining properties. First, it must drive particles either directly towards or directly away from a fixed point in space, the center of force, which is often labeled O. In other words, a central force must act along the line joining O with the present position of the particle. Second, a conservative central force depends only on the distance "r" between O and the moving particle; it does not depend explicitly on time or other descriptors of position. This two-fold definition may be expressed mathematically as follows. The center of force O can be chosen as the origin of a coordinate system. The vector r joining O to the present position of the particle is known as the position vector. Therefore, a central force must have the mathematical form  where "r" is the vector magnitude |r| (the distance to the center of force) and r̂ = r/r is the corresponding unit vector. According to Newton's second law of motion, the central force F generates a parallel acceleration a scaled by the mass "m" of the particle  For attractive forces, "F(r)" is negative, because it works to reduce the distance "r" to the center.