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One of the formalisms in variational calculus and in particular a formulation of classical mechanics (and also a version for geometrical optics, with eikonal in the place of principal function) is Hamilton-Jacobi equation which just got an entry.
Eventually, I would like to transform somehow the entry classical mechanics. Namely if we fill the sections which are there written but empty, it will grow beyond usability. I think apart from introduction, the entry should have passage between various formalisms. But the details on each formalism could be better on the separate page. Now the bulk of the entry is Poisson formalism which should be I think a separate entry. But it is not easy to engineer a good plan for this yet so let us continue adding material and we can transform the overall logic later. In any case, Hamilton-Jacobi formalims should be on equal footing with Hamiltonian formalism, Lagrangean formalism, Poisson formalism, Newton formalism etc. and some exotic structures like Nambu mechanics and Routhians should be mentioned and linked, in my opinion.
Thanks!
I have edited the formatting slightly.
Yes, I agree that the topic cluster of entries on “classical mechanics” deserves to be improved much more.
But I have one request: in traditional texts the term “classical field theory” is often used to refer to the Lagrangian or action functional itself. This is a mistake which we should try not to propagate on the $n$Lab. In order to account for this I have started to use the word “prequantum field theory”, borrowing the word “prequantum” from geometric quantization.
There is prequantum field theory given by data such as Lagrangians. From there we have two or three routes: passing to the critical locus, this is classical mechanics. Or passing to the critical locus and then hbar-deforming. This is perturbative quantum field theory. And finally, not passing to the critical locus at all, but doing full quantization. This is quantum field theory.
This is not criticizing aything you wrote, but just a remark that I kept thinking of recently when working on related $n$Lab entries.
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