# 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 *x*_{0} before the device and another axis *x*_{1} after. The entry data *(x*_{0, }Θ_{0}) of a ray determine its exit data *(x*_{0},Θ_{1}).

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 (levi@math.psu.edu) is a professor of mathematics at the Pennsylvania State University.