Mathematics behind the elliptical orbits of planets
This course shows how to mathematically prove the well-known Kepler's laws. The mathematical equations are derived and solved step-by-step.
This course shows how to mathematically prove the well-known Kepler's laws. The mathematical equations are derived and solved step-by-step. First, the two-body problem is formulated by using Newton's laws of motion and the universal law of gravitation. Then, the equations are solved, still preserving the physical intuition behind them. Besides, the conservation of the angular momentum, energy of the planet will be motivated and important parameters of the orbit such as: perihelion, aphelion, period, are explicitly written in terms of the energy, angular momentum, mass of the planet, etc., in order to show how the former are affected by the latter.
Conservation of the earth's angular momentum
FREE PREVIEWConservation of the energy of the earth
Equations of motion derived from the Lagrangian
Integrating the equation of motion part 1
Integrating the equation of motion part 2
Integrating the equation of motion part 3
Equation of the trajectory in polar coordinates
Proof that the trajectory is an ellipse
Proof that the sun is located at one of the two foci of the ellipse
2nd Kepler's law
Animation of the first two Kepler laws
Period of the orbit (3rd Kepler's law)
Calculation of the integral which appeared in the formula of the period