Talk:Hamiltonian field theory

Should have been it's own article long ago

The analytical mechanics and especially classical field theory articles on WP are currently a mess. Anything similar (e.g. Lagrangians for systems of particles and Lagrangian densities for fields) are mushed into each other within the same articles. Things would be much easier to follow if there were separate articles on general analytical mechanics, general classical field theory, Lagrangian mechanics, Lagrangian field theory, Hamiltonian mechanics, and Hamiltonian field theory. The first two are separate but each article needs work. The rest do not have clean cut articles (I'll check their talk pages in detail later).

So, I started this article, as it was a redlink and thre doesn't seem to be anywhere where Hamiltonian field theory is plainly described. I know the equations are written clumsily using nabla and time partial derivatives instead of compact tensor index notation, and the text is very pedantic, but I want to be absolutely clear which derivatives are used. Most of the literature is not clear about overdots for partial time derivatives, and we should avoid abuses of notation. Both of these are in footnotes.

Much more to follow. M∧Ŝc²ħε И_τlk 14:44, 31 July 2015 (UTC)[reply]

To do list, or scrap the article

Plenty of things for this article:

Priority

Field theoretic Piosson brackets,
Phase space of fields
Derivation from principle of least action, and/or Lagrangian field equations
Examples (Newtonian gravitational field? Classical EM field? Nonlinear fields? others...)
Relativistic covariant generalization (we could merge covariant Hamiltonian field theory into this article)

Other useful bits

Continuity equation in terms of the Hamiltonian
Relation to the stress-energy tensor?

If anyone can find an article or articles which already has all these things in one place, then in principle this article should be scrapped. If the above content is scattered here, there, everywhere (which seems more likely) they should be merged into this article. M∧Ŝc²ħε И_τlk 15:38, 31 July 2015 (UTC)[reply]

Given no objections, I've completed the merge in. Klbrain (talk) 22:42, 24 October 2017 (UTC)[reply]

I am no expert on the subject, but it seems to me that in general a Hamiltonian density can be a function of the partial derivatives of position, just think of a Lagrangian density defined as

{\mathcal {L}}={\frac {1}{2}}\left(\partial _{t}\phi \right)^{2}-{\frac {1}{2}}(\nabla \phi )^{2}-V(\phi )

which gives a Hamiltonian density

{\mathcal {H}}={\frac {1}{2}}\pi ^{2}+{\frac {1}{2}}(\nabla \phi )^{2}+V(\phi )

. If I'm correct I suggest modifying the expressions of

{\mathcal {H}}(\phi _{i},\pi _{i},\mathbf {x} ,t)\to {\mathcal {H}}(\phi _{i},\pi _{i},\nabla \phi _{i},\mathbf {x} ,t)

in such a way as to make this dependency explicit Prolachidofo (talk) 18:48, 21 January 2025 (UTC)[reply]

Wrong use of variational derivative?

In the section "Equations of motion", the EOM are written as

${\dot {\phi }}_{i}={\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}},\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}},$

with the side note that the variational derivative is given by

${\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\nabla \cdot {\frac {\partial }{\partial (\nabla \phi _{i})}}.$

I believe this is incorrect. A variational derivative should act on a functional, $H$ , not a function, ${\mathcal {H}}$ . This means the EOM should be written as

${\dot {\phi }}_{i}={\frac {\delta H}{\delta \pi _{i}}},\quad {\dot {\pi }}_{i}=-{\frac {\delta H}{\delta \phi _{i}}},$

and the sidenote should be written as

${\frac {\delta H}{\delta \phi _{i}}}={\frac {\partial {\mathcal {H}}}{\partial \phi _{i}}}-\nabla \cdot {\frac {\partial {\mathcal {H}}}{\partial (\nabla \phi _{i})}}.$

The variational derivative is often explained with a bit of handwaviness in physics (I know from experience) and while it is often didactally justified, it can be quite confusing sometimes. I think this section only adds to the confusion and so it must be changed. Thegozer100 (talk) 08:40, 17 March 2025 (UTC)[reply]

Why square brackets for poisson bracket?

In the section "Poisson bracket", square brackets [] are used for the poisson bracket. The wikipedia page on poisson brackets, as well as the source^* use curly braces {} for the poisson bracket. Are square brackets used in any source for the poisson bracket?

I don't have the book (Greiner & Reinhardt 1996), but I asked chat gpt which brackets the book uses.

I posted two topics here. I don't know when it is okay to change something because I'm new on Wikipedia. If I changed them too soon please let me know. Thegozer100 (talk) 08:50, 17 March 2025 (UTC)[reply]