Rough (one pass by a non native speaker of either language) translation of:
Connaissez-vous le pendule ?
by Alain Chenciner
I've used * as a place holder for formulas or references I didn't want to type.
------
Do you know the pendulum?
The movements of a pendulum of length l and mass m under constant gravity g (Figure 1.1) are ruled
by the differential equation *, that is to say it's ruled by the orthogonal projection onto the tangent to the circle at the point parametrized by the angle x of the equation * (the component normal to the circle is exactly compensated by the reaction of the bar, rigid but non massive, which holds the point mass m). We have used a point, as the physicist do, to denote differentiation with respect to time. Fixing * we obtain * or *.
Considered as a vector field on the plane with coordinates (x,y), the second member * of this equation has the phase portrait described by Figure 1.2.
This phase portrait only gives the geometry and orientation of the integral curves, ignoring their parametrization.
The phase portrait can also be obtained as the family of curves which are level sets of the energy function: *,
without much effort after local analysis (aided by the Morse Lemma) the singularities of this function, that is points where the partial derivatives are zero, it is not a submersion. In a neighborhood of points y = 0 and x multiples of 2pi, the quadratic form of second derivatives is positive definite and equilibrium is stable; in a neighborhood of points y=0 with x equal to pi plus an integer multiple of 2pi, the form is non-definite and equilibrium is unstable.
Well understood, the "real" phase space of the pendulum is the cylinder *, that is the unit tangent bundle * to the circle * (set of position-velocity pairs). The figure 1.3 gives a representation though one might prefer the one given by figure 7b of my article *. However, the phase portrait on * - that is after a "double covering" - which is the right object of study from the point of view of analysis (Figure 1.4).
A first level of comprehension of this assertion is obtained directly in the real domain : embed * in R^3 by the mapping *, and set *, noticing that we always have *. The curve (not conected if *) of energy H in * is identified with the intersection of two cylinders with equations * and * respectively.
These cylinders are represented by the figure 1.5 in the case where *. Let's integrate the pendulum equation. With position variable z, the energy relation is described by *.
The solution z(t) = *, verifies * and therefore the reciprocal function t(z) is defined by *,
that is to say if k < 1, the product of by k of Jacobi's elliptic function *.
One can deduce (see *) that * and *.
If k > 1, we find *, which describes the homoclinic orbit joining in infinite time the unstable equilibrium point to itself.
The reader will not resist, I hope, the comparison of the parametrization of the constant energy curves of the pendulum by the three Jacobi elliptic functions with the parametrization of the constant energy curves of the harmonic oscilator * by the circular (trigonometric) functions.
However, these formulas do not obtain there true stature until they are extended to the complex plane. To understand this, we recall the nature of the function * of a complex variable u. And, since this function degenerates to sin(u) when k goes to 0, we start with the case of the harmonic oscilator, that is to say the sine function and its (multivalued) inverse *.
The figure 1.6 anaylzes, in the real domain, the re-writting of *, as the composite funciton of * with the projection *.
The figure 1.7 does the same in the complex domain. Since we are unable to draw in R^4, we represent the cylinder which is the image of the first mapping with two half-lines of self-intersection, each one corresponding to a generator of the cylinder folding onto itself in R^3, and a ramification point of order two for the projection of the cyclinder in the complex plane z. After completion by two points at infinity, this cylinder becomes the Riemann surface (sphere) of the function *. In short, we have represented the ellipses and hyperbolas which are homofocal, images of the lines * and *.
The figure 1.8, where we have done the same thing for the function z = sn(u,k), shows the the cylinder is replaced by a torus from which two points have been removed (the poles of sn(u,k)) and therefore the Riemann surface of the function * is a torus. We reobtain figure 1.7 letting k go to 0.
In the doubled covering space *, the complex integral curve of energy *, intersection of two cylinders * in C^3 = R^6, is therefore diffeomorphic to a torus from which two points have been removed, therefore in the "true" complex phase space *, it's a torus minus a single point.
Remark. The expression for the period T(k) of a full oscilation of the pendulum can be deduced from the preceeding: if k < 1, for example, one obtains *, which is the real period of the function sn(u,k) which varies from 2pi (in the limit case of the harmonic oscilator, small oscilations) to infinity (in the limit case of the homoclinic orbit joining the unstable equilibrium to itself). But the function sn(u,k) also posseses a purely imaginary period, classicaly denoted by 2iK'(k) (see *). It was Paul Appel who, in 1878, gave, only in the case k< 1, the dynamical interpretation of this last period; he remarked that replacing the real time t by complex time it is the same as multiplying velocities by i and accelerations by i^2 = -1, that is to reverse the orientation of gravity. He then deduces immediately that the complex period obtained by leaving the pendulum at rest at position x is equal to the real period corresponding to the position pi - x (figure 1.9).
Guide for interpreting figures 1.7 and 1.8. These figures are obtained simply after the following remarks: in both cases we are representing a curve in C^2 (that is a real surface in R^4) given by an equation of the form w^2 = P(z), where P is a polynomial of degree 2 or 4 whose roots z_i are all real. We can consider this surface as the graph of the multivalued function *. The problem occurs at the level of points (z_i,0) where the restriction to this surface of the projection * is ramified, exactly as happens at (0,0) for the graph of the function *. Classically we represent such a surface as follows : we cut the plane in a certain number of places in the complement of which it is possible to choose a continuous square root of P(z). The surface now consists of pasting copies of this complement along the boundary of the cuts, and the only difficulty is to understand the pasting. In order to respect the change of determinations of * when z follows a small loop around a root, the pasting must associate one boundary in a leaf to the opposite boundary in another. The figure 1.10 indicates two perspectives of the representation in R^3 of the graph of * while figure 1.11 shows the pasting of the two leaves for the graph of *.
Remarks. 1 - One has a certain amount of latitude in the choice of cuts and in the representation in R^3. For example, if, in the respresentation of sine, we prioritize the imaginary part of w more than its real part and replace the two cuts by a single cut along [-1,1], the figure 1.7 turns into figure 1.12 :
2 - For the mappings from R^2 to R^3 depicted in 1.7, 1.8, 1.10, and 1.12 the bifurcation point have an unavoidable (stable) singularity called a Whitney umbrella, where the simplest local model is *.
This singularity is also found in the cross-cap, representation of the MÃ¶ebius band where a neighborhood of the boundary is embedded as an standard annulus in R^3, which allows one to paste in a disk forming the simplest model for the projective plane and more generally any non-orientable surface (see the figures in *).
Exercise: Draw the analogous figures for a general polynomial P (hyperelliptic integrals) and show that the Riemann surface of the function * is an orientable surface with genus g if the polynomial has degree 2g+1 or 2g+2 (see for example *).
Some disappointed remarks: Classical mechanics is no longer fasionable for (certain) mathematicians. At the university of Paris VII for example, where after some years my proposals to teach a yearly course on geometry and mechanics for masters students are faced with claims of being impractical by certain responsible collegues. Too hard for the students, they say. And I can only agree with them. What need is there to distinguish Lagrangian and Hamiltonian, tangent and cotangent, while we take care in eliminating from the curricula the notion of duality itself. Besides, there are practically no longer any students to chose the geometry course. Hello, hello, don't hang up....