# Is symplectic form in Hamiltonian mechanics a physical quantity?

Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure.

The real reason why I'm asking lies not in mechanics, but in GENERIC. There apart of Poisson bracket the second bracket is added. This bracket is responsible for the irreversible evolution and depends highly on a physical system in hand. While I was musing about it I wondered whether I can attribute these two brackets as physical quantities and if no, how shall I at least call them as opposed to physical quantities?

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I would say no, the symplectic form isn't a physical quantity. It's rather a quantity specific to the phase space formulation of a physical system. If you choose not to formulate things in phase space, the symplectic form is absent.

Moreover, even if you do work in phase space, in a system with constraints, the physical quantities are really only defined on the space of gauge orbits in the constraint submanifold. The symplectic form doesn't live there, but rather lives in the full phase space.

Yet another reason for arguing for non physicality is that all symplectic manifolds of the same dimension are locally identical (Darboux theorem) - the symplectic form is the same. However not all physical systems of the same dimension are the same.

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What makes a quantity physical? –  NikolajK Apr 28 '13 at 13:58
@NickKidman I would refrain from answering such question however directly related to my question it is. –  Yrogirg Apr 28 '13 at 14:07