Kepler’s Laws from Comoment Maps

2.2 Energy and Conserved Vectors

Definition 3 Energy, angular momentum, and Laplace–Runge–Lenz vector
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Uses

Lean declarations

For a phase point \(z = (q, p)\), define:

\[ H(q,p) = \frac{\| p\| ^2}{2m} - \frac{\mu }{\| q\| }, \qquad L = q \times p, \qquad A = p \times L - \frac{m\mu }{\| q\| }q. \]

Here \(H\) is the total energy (kinetic minus potential), \(L\) is the angular momentum vector, and \(A\) is the Laplace–Runge–Lenz vector. All three are proved constant along every non-collision solution in Chapter 4.