This constant areal velocity can be calculated as follows.

Kepler's second law implies that the areal velocity is a constant of motion.

The constancy of areal velocity may be illustrated by uniform circular and linear motion.

See the article on areal velocity.

For a single particle, areal velocity provides a geometrical interpretation of angular momentum.

The areal velocity of the tangent vector is:

Thus, the areal velocity is constant for a particle acted upon by any type of central force; this is Kepler's second law.

Now apply the Frenet-Serret theorem to find the areal velocity components:

The concept of areal velocity is closely linked historically with the concept of angular momentum.

The areal velocity vector is perpendicular to this surface, and, in general, varies in both magnitude and direction.

Angular momentum is conserved if and only if the areal velocity is a constant vector.

The direction of the angular momentum vector L is always the same as that of the areal velocity vector.

In planar problems, such as the orbit of a planet about the sun, the direction of the areal velocity vector is perpendicular to the orbital plane.

Since the speed v is likewise unchanging, the areal velocity vr is a constant of motion; the particle sweeps out equal areas in equal times.

Areal velocity (sector velocity, sectorial velocity) is the rate at which area is swept out by a particle as it moves along a curve.

The areal velocity vector can be placed at the moving point B. As the particle moves along its path in space, it sweeps out a cone-shaped surface.

In some problems, a component of angular momentum is conserved, and in these cases, the corresponding component of areal velocity is constant.

Patrick d'Arcy announces discovery of the principle of angular momentum in a form known as "the principle of areas" (areal velocity).

The magnitude of h also equals twice the areal velocity, which is the rate at which area is being swept out by the particle relative to the center.

In differential geometry, especially the theory of space curves, the Darboux vector is the areal velocity vector of the Frenet frame of a space curve.

Thus, in the Kepler problem, the angular momentum of a planet about the sun is conserved and the areal velocity is a vector of constant magnitude perpendicular to the orbital plane.

As it does so, the object's motion will be described by two vectors: a translation vector, and a rotation vector ω, which is an areal velocity vector: the Darboux vector.

To derive this expression we need the specific relative angular momentum, h, which is related to the areal velocity and a constant of the motion of a planet and the formulae for free orbits.

J. Casey, "Areal Velocity and Angular Momentum for Non-Planar Problems in Particle Mechanics," American Journal of Physics, Vol.

Conversely, if the motion under a conservative force F is planar and has constant areal velocity for all initial conditions of the radius r and velocity v, then the azimuthal acceleration a is always zero.