Why are integrals over cycles called periods?

Question

In the statement of de Rham theorem, a pairing is defined as follows:

$H_i(X, \mathbb R) \times H^i_{\mathrm{de Rham}}(X) \rightarrow \mathbb R$

It is given by

$\left( \left( \sum a_i \gamma_i \right) , \omega \right) \mapsto \sum a_i \int_{\gamma_i} \omega $.

Here $a_i$ takes real values, and $\gamma_i$ are integral homology cycles, and $\omega$ is an $i$-form.

The integral given above is called the period of the above integral, and the isomorphism given by this pairing is sometimes called the period isomorphism.

Question:

Why is the above integral of a closed form over a cycle called a period?

My peeve with this terminology of "period" is that it does not agree at all with anything else I know about this word. They are the following: 1) The period of a periodic function. 2) Periods as generalizations of algebraic numbers, as integrals of algebraic(rational) expressions over domains in Euclidean spaces defined by algebraic inequalities.

Indeed, this strange use of the word "period" is used even by Ahlfors in his book on complex analysis, for integrals of holomorphic(rather, meromorphic? depending on the domain ...) functions over loops.

I do not understand why on earth the word period appears in this setting of integrating on abstract manifolds. True, the cycles capture some underlying geometry of the space. But why is something a "period" when you integrate a form over a cycle? Why is the integral of a $1$-form over a line segment not a period?(Or is it also period, in some definition I am not aware of?)

Answer 1

In the case of elliptic curves, integrating some fixed holomorphic differential over the first homology gives the lattice of periods of the corresponding elliptic functions. (These elliptic functions are certain meromorphic functions on $\mathbb C$ which are "doubly periodic", i.e. are invariant under $z \mapsto z + \omega$, where $\omega$ lies in the lattice of periods obtained as above.)

This is the source of the terminology.

Answer 2

I must admit that this is the first time I heard the suggestion that the word "period" came into complex analysis/algebraic geometry from the period of the pendulum. The word was certainly used in astronomy before the theory of the pendulum was known -- e.g. Kepler stated his third law in 1618 as "The squares of the periodic times are to each other as the cubes of the mean distances" -- and uniform circular motion gives a "period" that was known before the period of the pendulum.

In any case, the concept of "period" is not a big deal until one discovers functions with two (or more) periods, which first happened when Gauss discovered this property of the lemniscate sine function in 1797. The lemniscatic sine, $sl(u)$,is the inverse function of the elliptic integral $ u=\int^{x}_{0} \frac{dt}{\sqrt{1-t^4}}, $ and Gauss called its real period $2\varpi$, using the variant form of the Greek letter pi. Presumably this was to stress its analogy with the period $2\pi$ of the sine function.

Answer 3

I believe that as sigfpe points out in the comments, the etymology comes from the sort of integrals which appear when one computes the periods of oscillation of mechanical systems.

Indeed, consider a particle of unit mass moving in the real line under the influence of a potential $V(x)$. The phase space of this system is the cotangent bundle of the real line, which we can identify with $\mathbb{R}^2$. The phase space is foliated by the physical trajectories, which in this simple example are labelled by the energy $E$ of the trajectory. Suppose that $E$ is such that trajectories are closed. The following picture illustrates the situation.

Potential function http://dl.dropbox.com/u/5096148/Potential.png

If the particle lies in the interval $[a,b]$ it will remain in that interval for all time and its motion will be periodic with period given by the (improper) integral $$ T = \sqrt{2} \int_a^b \frac{dx}{\sqrt{E-V(x)}}.$$

Here the cycle is the one-dimensional submanifold of the phase space (with coordinates $(x,y)$) given by the equation $$\frac12 y^2 = E - V(x),$$ and the period is the integral of the differential $dx/y$ on the cycle.

A typical example is that of a simple pendulum, where $V(x) = g\ell (1-\cos x)$, where $\ell$ is the length of the pendulum and $g$ the acceleration due to gravity. If we let $E = g\ell (1-\cos x_0)$ then the period of oscillation becomes $$ T = 2 \sqrt{\frac{2\ell}{g}} \int_0^{x_0} \frac{dx}{\sqrt{\cos x - \cos x_0}}.$$

To turn this into an elliptic integral we change variables to $\theta$ defined by $$ \sin\theta = \frac{\sin x/2}{\sin x_0/2} $$ in terms of which the period integral becomes an elliptic integral of the first kind $$ T = 4 \sqrt{\frac{\ell}{g}} \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-\sin^2(x_0/2)\sin^2\theta}}.$$

I'm not sure of dates, but I would be surprised if this (which was certainly known to the Bernoullis) did not predate the uses of period in algebraic geometry.