Newton's laws of motion Newton in his "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), first published in 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. A fourth law is often also described in the bibliography, which states that forces add up like vectors, that is, that forces obey the principle of superposition. Newton's laws are applied to objects which are idealised as single point masses,
Kepler's laws of planetary motion that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law. The mathematical model of the kinematics of a planet subject to