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This is the Forum you wrote about? :-) The Hamiltonian (or Lagrangian) has in most cases no explicit notion of time, but "generalized" impulses: interdependencies of variables and the corresponding changes of them. In a sense, the notion of invariance as used in AMMs is just one half of a full Hamiltonian: the dynamics is missing. Hamiltonian mechanics can (in most cases) be easily transformed into Lagrangian and vice versa: a so called "Legendre transformation". Some things are easier in Hamilton formalism, but difficult in Lagrange, for other it is the opposite. With best regards, |
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It'd be interesting to know if the math behind CFMMs can be structured in a Hamiltonian/Lagrangian framework? (My physics background is far enough in my past I don't know which of these would make more sense...)
There is no useful sense of time in the fundamental CFMM math, but I wonder if replacing this with the numeraire makes sense? If R is the quantity of an asset and Q is the quantity of the numeraire, derivatives dR/dQ already play a fundamental role in the AMM (as spot prices)... and R * dQ/dR also plays a fundamental role (total value of the asset in the system)...
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