https://play.3speak.tv/embed?v=mes/8niolny6
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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 properties – 36:32
- The derivative of v x h = a x h = GM u' – 40:39
- v x h = GM u + a constant vector c – 45:49
- v x h, u, and c are on the xy-plane – 47:35
- Choose c to be along the standard basis vector i and get the polar coordinates of the position vector – 53:22
- The dot product between r and v x h yields a formula containing polar coordinates r and θ – 55:54
- Evaluate the formula via the scalar triple product to obtain the polar equation of an ellipse! – 1:01:50
- Compare with my earlier video on the polar equation of an ellipse via the unified conic theorem where the eccentricity is constant and is defined as the ratio of the distance to the focus divided by the distance to the directrix – 1:08:11
- The orbit of a planet is a closed curve, so the conic has to be an ellipse, this proves Kepler's first law – 1:09:41
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