page_title: Angular momentum
section_title: Newton's derivation
hierarchical_section_title: Angular momentum / History / The Law of Areas / Newton's derivation
caption_attribution_description: English: Newton's derivation of the area law from the laws of motion: as an object orbits the Sun it sweeps out equal areas in equal intervals of time. An object (not shown) is in orbit around the Sun (Point S). The effect of the Sun's gravitational attraction is represented as a sequence of impulses, always directed towards point S. In the limit of infinitely many small impulses the representation approaches infinitely close to smooth gravitational attraction. When the object is at point B it receives an impulse towards point S. Without that impulse the object would proceed to point c. The actual displacement BC follows from the rules for velocity composition; the displacement BC is the vector sum of the displacements BV and Bc. The triangles SBc and SBC have the same base and the same height, hence they have the same surface area. At point C the object receives another impulse towards point S. Without that impulse the object would have proceeded to point d in an equal interval of time. The impulse towards point S makes the object proceed to point D. The triangles SBc, SBC, SCd, SCD, SDe, SDE all have the same surface area. In the limit of the time intervals going to infinitisimally small the sequence of lines connecting the points B, C, D, E, etc approaches infinitely close to the actual smooth trajectory. العربية: إثبات نيوتن الهندسي لقانون المساحات
context_page_description: In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant.
  In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector; the latter is p = mv in Newtonian mechanics. This definition can be applied to each point in continua like solids or fluids, or physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it.
  Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and orbital angular momentum. The spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin. The total angular momentum of an object is the sum of the spin and orbital angular momenta.
context_section_description: As a planet orbits the Sun, the line between the Sun and the planet sweeps out equal areas in equal intervals of time. This had been known since Kepler expounded his second law of planetary motion. Newton derived a unique geometric proof, and went on to show that the attractive force of the Sun's gravity was the cause of all of Kepler's laws.
  During the first interval of time, an object is in motion from point A to point B. Undisturbed, it would continue to point c during the second interval. When the object arrives at B, it receives an impulse directed toward point S. The impulse gives it a small added velocity toward S, such that if this were its only velocity, it would move from B to V during the second interval. By the rules of velocity composition, these two velocities add, and point C is found by construction of parallelogram BcCV. Thus the object's path is deflected by the impulse so that it arrives at point C at the end of the second interval. Because the triangles SBc and SBC have the same base SB and the same height Bc or VC, they have the same area. By symmetry, triangle SBc also has the same area as triangle SAB, therefore the object has swept out equal areas SAB and SBC in equal times.
  At point C, the object receives another impulse toward S, again deflecting its path during the third interval from d to D. Thus it continues to E and beyond, the triangles SAB, SBc, SBC, SCd, SCD, SDe, SDE all having the same area. Allowing the time intervals to become ever smaller, the path ABCDE approaches indefinitely close to a continuous curve.
  Note that because this derivation is geometric, and no specific force is applied, it proves a more general law than Kepler's second law of planetary motion. It shows that the Law of Areas applies to any central force, attractive or repulsive, continuous or non-continuous, or zero.
caption: Newton's derivation of the area law using geometric means.