Telescopes and Symplectic Mappings
By Mark Levi
Symplectic geometry, tracing its origins to the work of Poincaré on Hamiltonian systems, and currently a very active field, reached a high level of abstraction. Here is a simple concrete example where "symplectic" approach predicts and explains the following physical fact:
Any optical device (e.g. a telescope) which converts a parallel beam to a narrower parallel beam must necessarily magnify objects.
This, as I will show next, is a manifestation of the obvious fact that if an area preserving a map squeezes in one direction, it must expand in another.
Figure 1. We place an axis x0 before the device and another axis x1 after. The entry data (x0, Θ0) of a ray determine its exit data (x0,Θ1).
Any optical device (in two dimensions)—schematically, the black box in Figure 1—gives rise to a map which assigns to each ray’s entry data (\(x_0\),\(y_0\)) where \(y_0\)=\(sin \: \theta_0\), the corresponding exit data (\(x_1,y_1\)). Parallel beams, e.g. \(CD\) and \(C'D'\) in Figure 2, correspond to horizontal segments in the \(xy\)-plane. Horizontal segments map under \(\varphi\) to horizontal segments; moreover, \(\varphi\) shortens these segments since the device narrows parallel beams, as in Figure 2.
Figure 2. Left: Beam CD exits as beam C’D’. Right: φ(CD) = C’D’.
Now \(\varphi\) is area-preserving. And since \(\varphi\) squeezes the rectangle \(ABDC\) (Figure 3) in the \(x\)-direction, it must stretch in the \(y\)-direction. This \(y\)-stretching means that the angles between parallel beams are magnified. But this is precisely what the optical magnification of objects amounts to. For example, the reason a telescope allows us to tell that a distant speck is actually a ship and not a dot is that it increases the angle between two beams, one from the stern and the other from the bow, thus making these beams fall onto different "pixels" on our retina.
Figure 3. Narrowing of the beams causes widening of the angles between beams, i.e. the optical magnification.
The proof of area-preservation in the footnote, due to Poincaré, admits a "hands-on" palpable mechanical interpretation, as described in [2]. More on symplectic maps and lenses can be found in the remarkable book [1]. And there are interesting open questions that we do not address here on the relationship between recent results of symplectic geometry and optics.
To maximize simplicity, I minimize the dimension to two.
To see why, consider the travel time T (\(x_0,y_0\)) (called the optimal distance), and note that y\(_0 = -T_{x_0} (x_0, x_1)\), \(y_1 = T_{x_1} (x_0, x_1)\), subscripts denoting partial differentiation. Then for a closed curve \(\gamma_0\) in the (\(x_0\),\(y_0\)) plane, parameterized by s \(\in\) [0,1] we get \(=\int_0^1 \frac{d} {ds}T(x_0(s), x_1(s))ds = \int_0^1 (-y_0 x_0^\prime + y_1 x_1^\prime)ds = \int_{\gamma_0} y dx + \int_{\varphi(\gamma_0)} y dx \).
Acknowledgments: The work from which these columns are drawn is funded by NSF grant DMS-1412542.
References
[1] Guillemin, V., & Sternberg, S. (1984 or later). Symplectic Techniques in Physics. Cambridge University Press.
[2] Levi, M. (2014, March). Classical Mechanics with Calculus of Variations and Optimal Control: an Intuitive Introduction. AMS, 46 & 275.
Mark Levi ([email protected]) is a professor of mathematics at the Pennsylvania State University.