Lagrange

Annotated, from The Principia, Section II Proposition 1 Theorem 1

Calculus of Variations

Celestial Mechanics

Kepler's Laws

Elliptical Areas

The sectorial velocity.

Kepler's Second Law

Let a body be launched at point A towards point B, subject to gravitational attraction towards S. The force is always in the direction of S, but otherwise unknown in magnitude. Newton's first law, the parallelogram rule on compound motion, and Euclidian geometry are applied to show that the body will sweep out in continuous motion equal areas in equal times.

The unknown positions of the body at equal intervals DT of time are shown in capital letters ABCD... The area of triangle SAB is the same as that of SBc because by Newton's first law bases AB = Bc over equal times DT, and the corresponding triangles have the same altitude (shown annotated). The motion of the body from B to C is found by the parallelogram rule: Add as shown the inertial motion Bc to the unknown impulsive motion Cc caused by the impulse at B, which motion must be in the direction BS. BS is parallel to Cc so that the altitude (shown annotated) of triangle SBC is the same as that of SBc. Therefore

area SAB = area SBC

similarly area SBC = area SCD etc.

The construction is independent of the magnitude of DT. In the limit of small DT the impulsive force on the body over the path ABCD.... is continuous. The time elapsed over this interval is the integration of DT. The area swept is the integration of the constant dA, the differential areas, over the same interval, and the total area swept is proportional to this interval. This is Kepler's second law.

WT presents the vertices of the white triangle and its area dA. The sectorial velocity dA/DT is also computed.

YT presents the vertices of the yellow triangle and its area dA. The sectorial velocity dA/DT is also computed.

The gravitational impulse is shown in green, The application of the parallelogram rule to this impulse is shown in blue.

F=ma presents the incremental change in velocity vector due to the impulsive central force, both components.

Sectorial Velocity

Sectorial velocity defined in Landau and Lifschitz is the quotient of the area swept by the planet between impulses of gravitational attraction and the differential of time between those impulses. In these examples the area swept during a time interval is presented in the lower right, with the computed sectorial velocity. This is compared to the quotient of angular momentum and mass, which is shown to be twice the sectorial velocity for all Keplerian systems.

The GPS is in circular orbit at 2.661e7 m of mass 1000 kg. In the lower right the area of the swept yellow triangle is of area 1.26e13 m^2 in the time step 245 sec.

Stationary Action

Moon Surface

Asteroid Eros