An elementary proof of Kepler's first law, i.e., that bounded planetary orbits are elliptical, is derived without the use of calculus. The proof is similar in spirit to previous derivations, in that conservation laws are used to obtain an expression for the planetary orbit, which is then compared to an equation for an ellipse. However, we derive the latter equation using trigonometry and two well-known properties of the ellipse. Calculus is avoided altogether.
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The Liouville–Arnold integrability theorem for Hamiltonian systems relies on finding as many independent constants of motion in involution as there are degrees of freedom in the system.
7.
A formulation using relative coordinates and reduced mass may be followed to overcome this approximation if needed.
8.
These two properties are not independent. The second may be derived from the first, for example, by means of Fermat's principle. However, using both of the properties allows us to simplify the algebra and avoid calculus.
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Our re-use of the symbols
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2021
Author(s)
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