# 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?

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.

• What makes a quantity physical? – Nikolaj-K 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

The Symplectic Form in Hamiltonian Optics

I'd like to answer this with a specific example from my own field, optics, where the symplectic form on optical phase space computes the Optical Invariant for a pair of rays, also called the Helmholtz invariant (or, commonly, the Lagrange Invariant, which is not a word I like, given this is a clearly Hamiltonian / Symplectic Geometry notion). In optics this is a hugely fundamental and useful notion.

More generally, the symplectic form $\omega(X,\,Y)$ evaluated for any pair of vectors $X,\,Y \in T_x(T^*(\mathcal{M}))$ in the tangent space at $x\in T^*(\mathcal{M})$ yields a conserved, measureable quantity whenever the vectors are Lie-dragged by a Hamiltonian flow, or more generally, pushed forward by a general symplectomorphism $\varphi: T^*(\mathcal{M})\to T^*(\mathcal{M})$, i.e. mapped by latter’s differential / Jacobi matrix; that is:

$$\omega_x(X,\,Y) = \omega_{\varphi(x)}(\mathrm{d}\varphi_x\,X,\,\mathrm{d}\varphi_x\,Y)\tag{1}$$

Conserved, measurable quantities are often useful in physics. Exactly how useful they are probably depends on the exact field under study, but, as I've said, this notion is highly useful in optics. The general idea of (1) - further to transport by a Hamiltonian flow - is important in optics, as often devices such as the abrupt dielectric interface of a lens impart a symplectomorphism that isn't physically the flow $\exp(X\cdot t)$ of a Hamiltonian vector field $X$ evaluated $\varphi=\exp(X\cdot 1)$ at $t = 1$, although one can often find an equivalent, exponentiated Hamiltonian vector field to represent these abrupt transformations. Another example is where one wants to think of an optical subsystem as a discrete, "black box" and not have to worry about the box's physical innards; there is then no physical Hamiltonian flow here either, although, again, we can represent our symplectomorphism in the form $\varphi=\exp(X\cdot 1)$ if needed for analysis.

Moreover, of course, the $N^{th}$ exterior power $\omega^N$ (with $N = \dim(\mathcal{M})$ is the dimension of the underlying configuration space $\mathcal{M}$) is also a conserved quantity when evaluated at $2\,N$ vectors which are pushed forward. So we have Liouville's theorem, that the phase space volume $\int_\mathcal{R}\mathrm{d}\omega^N$ of region $\mathcal{R}$ is conserved when so mapped. In optics, the volume $\int_\mathcal{R}\mathrm{d}(\omega\wedge\omega)$ filled by a system of rays is called the étendue, throughput or optical grasp (see footnote) of the light field that the rays represent.

Étendue and the Optical Invariant are often confused for one another in the literature. The fact that they are both conserved and approximately related to one another in very special, but useful, cases doesn't help to mitigate the confusion (the invariant squared is approximately proportional to the étendue for a circularly symmetric bundle of rays).

Applications

The usual application of the Symplectic Form / Optical Invariant $\omega$ is in conjunction with the linearization of the symplectomorphism $\varphi: T^*(\mathcal{M})\to T^*(\mathcal{M})$ defining the optical system or subsystem in question. This linearization is used to approximate the transformation wrought by the system on phase space points that are near a reference point, which is the so called "chief ray" in a ray bundle ("reference ray" is a term I like better - there’s nothing special about the "chief ray" aside from its being an arbitrarily chosen phase space point to serve as the origin to linearize the symplectomorphism around). The reference ray is often along the optical axis of a system but it does not have to be, if indeed the system has the rotational / reflexional symmetries needed to allow a meaningful definition of the optical axis - many of the big uses of the optical invariant and étendue are in non-imaging systems. The reference ray can be any ray through the system. Given the ray’s state $x_i\in T^*(\mathcal{M})$ at the system input, one calculates the ray’s state $x_o = \varphi(x_i)$ at the system output. The linearized transformation is then $\mathrm{d}\varphi_{x_i}:T_{x_i}(T^*(\mathcal{M})) \to T_{\varphi(x_i)}(T^*(\mathcal{M}))$. Naturally, $\mathrm{d} \varphi_{x_i}\in\mathrm{Sp}(4,\,\mathbb{R})$ and has a symplectic matrix. The Jacobian, i.e. the determinant of this matrix, is always unity (as it is for a symplectic operator of any dimension).

The linearized approximation then calculates the images of points nearby $x_i$ as $\varphi(x) \approx \varphi(x_i) + \mathrm{d}\varphi_{x_i}\,(x-x_i)$. In this use, it is a paraxial approximation, i.e. the calculation of the evolution of rays that are near to the reference (chief ray) in phase space.

So the optical invariant is a tangent space, paraxial concept, whereas étendue $\mathscr{S}=\int_\mathcal{R}\mathrm{d}(\omega\wedge\omega)$ and its conservation are non paraxial, nonlocal notions.

The conservation of the optical invariant yields deep insight into how optical systems work very simply. Before some examples, one must mention that the optical momentum is the transverse component (that normal to the reference ray) of the wavevector, therefore the linearized momentum components are $p_x = n\,\theta_x;\,p_y = n\,\theta_y$, where $n$ is the medium’s refractive index at $x_i$ and the $\theta_j$ are the angles made with the reference ray. A simple example now gives some idea of the notion’s power. Compute $\omega$ for two rays diverging at angle $\theta_i$ from a point laterally offset a distance $d$ from the reference ray. Now suppose the system has transverse magnification $M$, so that the rays converge at the output at a transverse distance $M\,d$ from the reference ray. Then the angle at which the rays converge is $\theta/M$ and so the system’s numerical aperture is always scaled by the reciprocal of the transverse magnification. An immediate consequence is that the $\text{resolution}\times\text{field of view}$ product is unchanged by an optical system. Not surprisingly, therefore, you can’t extract any more information from an image by magnifying it. Apparent information content gains by imaging systems only arise because the systems match light fields to whatever image acquisition technology - be it a CCD or a retina - is on hand and thus overcome technological limits.

Footnote

I’ve little idea where the term "optical grasp" comes from - I find it befuddling even as a native English speaker - but perhaps it has to do with the metaphorical idea of a hand in phase space holding a bundle of rays and the bigger the volume they take up in phase space, the wider the breadth (grasp) of the grip needed to hold them. It seems English is not alone in making up really unhelpful or arcane words for this notion - German has Lichtleitwert; something like "light conductance": go figure. Étendue ("spread") is for me by far the most evocative and helpful word, and its a shame that there are no good translations in English (or German, as far as I can tell).