If we ask whether we can use this method for generalised coordinates the answer is yes. As long as we can perform general linear transformations on those coordinates.
So this is not dependent upon the type of coordinate system used but only upon the ability to perform linear transformations.
We, therefore, have a unique presentation of the Hamiltonian. We can try to further generalise this idea by guessing that the radii are components of a metric as shown below.
This gives something that looks, perhaps, background free. Does this suggest that the Hamiltonian must then become a matrix of some sort? Perhaps the presence of the exponent implies this as well as the fact that the radii are components of the metric which is a higher structure, a tensor.
Overall...to where does this lead us. What physical scenarios might make use of all this that we have been suggesting?
See Part II: First Adventure and Physical Reality.
Notice there is, if you squint, an analogy between this Hamltonian and that used to define the action S = integral _/L# where L is the Lagrangian, _/is the integral and # is a p-form.
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