Many thanks, this example was given in some lecture notes that can be found by googling “Hamilton-Jacobi equation” to find a site by Michael Fowler from physics at the University of Virginia. He then goes on to discuss the central force problem, which I amplified in the notes. The advantages of the Hamilton dynamics and the Hamilton Jacobi dynamics are not clear at first, but once they are understood they give a lot of mileage as you have just shown. It should be possible to demonstrate a precessing orbit from the Hamilton equations.

426(3).pdf

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