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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?)

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I thought it came from the theory of elliptic integrals / functions but I don't have a reference. –  j.c. May 20 '10 at 14:32
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Long story short, it comes from the case where your variety is an elliptic curve. A function with source an elliptic curve can be thought as a doubly periodic function on $\mathbb{C}$, and the periods become... well, periods. That is, the segment joining the origin and a (fundamental) period is one of the generators of the first homology of the elliptic curve. –  Andrea Ferretti May 20 '10 at 15:56
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And I always thought it was because the integrals you use to compute the periods of an elliptic curve were the same as those used to compute the period of a pendulum, after some suitable change of variable. –  Dan Piponi May 20 '10 at 16:27

3 Answers 3

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.

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I simply used the pendulum as an example, to tie this with the elliptic integrals mentioned elsewhere in this thread. My point was that this stems from the notion of periodic motion in mechanical systems. The Kepler problem is one such system and can be treated analogously to the pendulum once conservation of angular momentum is used to arrive at an effective one-dimensional system. Of course, the "modern" way to do Kepler is to exploit the additional symmetry provided by the Laplace-Runge-Lenz vector. The problem then becomes algebraic without the need to integrate. –  José Figueroa-O'Farrill May 21 '10 at 9:32
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Fair enough. I like the pendulum illustration, but I think that there are better bets for the historical origins of the term "period". We know that periodicity was observed in planetary motion before the motion of the pendulum was understood properly, and indeed elliptic integrals arose with the arc length of the ellipse (Wallis 1655) before they arose with the pendulum. –  John Stillwell May 21 '10 at 12:42
    
As I said, the pendulum is merely the most easily accessible (to me, at least) example of a mechanical system whose period is given by an elliptic integral. I would agree with you that planetary motion was understood before the pendulum, and taking that as the origin of the term 'period' does not contradict my belief that the term derives from the periodicity of motion. I insist, though, that I cannot back this up with a reference. –  José Figueroa-O'Farrill May 21 '10 at 17:09

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

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.

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+1 for the pretty fonts! –  Mariano Suárez-Alvarez May 21 '10 at 3:03
    
Do you have a reference to back up your belief? See John Stillwell's answer which seems to contradict it. –  Victor Protsak May 21 '10 at 6:58
    
Mariano, thanks: <code>\usepackage{concrete}<\code> :) –  José Figueroa-O'Farrill May 21 '10 at 9:27
    
Victor: no, I'm afraid I don't have a reference. I would have cited it otherwise. –  José Figueroa-O'Farrill May 21 '10 at 9:28

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.

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Are you sure about this? I think that sigfpe is right in his comment and that the etymology comes from the fact that elliptic integrals compute the period of oscillation of a pendulum. –  José Figueroa-O'Farrill May 20 '10 at 18:08
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A thought: this pendulum business might be the reason periodic functions are thus named since in both cases things repeat, well, periodically. It's not so clear that you can disentangle the etymology so cleanly here. Elliptic functions are higher-dimensional analogues of trigonometric functions, the the periodicity of trigonometric functions is closely related to periodicity of pendulums. –  KConrad May 20 '10 at 20:43
    
For the purpose of etymology and history, I would phrase this differently. Integrating $dz/z$, indefinitely, or over paths that need not be closed, gives the logarithm, the inverse of a periodic function. The periods of the exponential correspond to the indeterminancy of the logarithm. That, in turn, corresponds to the indeterminancy of the path of integration. The indeterminancy of paths are homology classes. Thus integrating the form over homology classes gives the periods of the inverse of the integral. (this is all the same with elliptic curves and functions in place of the exponential) –  Ben Wieland May 21 '10 at 0:38

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