Kepler's First Law: Planets Revolve around the Sun in an Ellipse

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In this video, I derive Kepler's 1st law of planetary motion entirely from Newton's second law of motion and Newton's law of universal gravitation, while making extensive use of the calculus of vector functions, to prove that a planet orbits the Sun in an elliptical orbit where the Sun is a focus. This extensive proof demonstrates the power that calculus is in explaining observational astronomical data directly from the laws of physics.

math vectors calculus ellipse keplerslaws

Timestamps

- Kepler's Laws of Planetary Motion – 0:00 - Proof of Kepler's first law: Orbits around the Sun are ellipses – 2:51 - Recap on ellipses: The sum of distances from a point to two fixed points called foci is constant – 3:08 - Newton's second law of motion: The net force on a body is equal to the mass times acceleration – 7:34 - Newton's law of universal gravitation: Masses attract each other proportional to their masses and inversely proportional to the square of the distance between them – 8:22 - Equate the two laws to get the formula for the acceleration – 12:35 - The derivative of the cross product between position and velocity vectors is zero, so r x v = a constant vector h, thus the orbit is one in a plane – 16:49 - Rewrite the constant vector h using the position unit vector u – 22:18 - Evaluate a x h to get a vector triple product – 28:57 - Recap on the dot product and its properti…

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