# Mathematics behind the elliptical orbits of planets

Why planets move in elliptical orbits

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.

What you’ll learn

• The mathematical reason why planets move in elliptical orbits.
• How to prove the three Kepler’s laws.
• The physical quantities which influence the perihelion, aphelion, eccentricity of the orbit.
• How to calculate the period of the orbit.

Course Content

• Formulation of the problem and Equations of motion –> 2 lectures • 23min.
• Conservation of the energy and angular momentum of the planet –> 2 lectures • 16min.
• Equations of motion derived from the Lagrangian –> 1 lecture • 14min.
• Solving the equation of motion –> 3 lectures • 56min.
• Equation of the trajectory and Kepler’s laws –> 7 lectures • 1hr 33min.

Requirements

• Calculus (especially: derivatives, integrals).
• Not necessary, but knowledge of Lagrangian mechanics might help (it is used in just one video of the course).
• Newton’s laws of motion and universal law of gravitation.

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.

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