Two Action-Angle Surprises on the Sphere

Abstract

The linearized Poincaré map of a periodic orbit of a completely integrable Hamiltonian system is examined in the light of the finer description we get by using coordinate changes in the Lagrangian odd symplectic group. In particular, we obtain non-eigenvalue invariants called moduli. These invariants are surprisingly subtle to calculate even in the case of the geodesic flow on a 2-sphere, and reveal dynamic-geometric information that is otherwise symplectically invisible.

Appendix

As we do not know the Hamiltonian H as an explicit function of the actions $j_1$ and $j_2$ we compute the necessary partial derivatives of H as follows. Since

$$\begin{aligned} j_1 = J_1(h,j_2) = \frac{1}{2\pi } \, \int _{(H_0)^{-1}(h)} p_s\, ds = \frac{1}{2\pi } \, \int _{(H_0)^{-1}(h)}\left( 2h-\frac{1}{f^2}j_2^2\right) ^{1/2} \, ds \end{aligned}$$

is a period integral, we may differentiate under the integral sign to obtain

$$\begin{aligned} \left. \frac{\partial J_1}{\partial h} \right| _{j_2=0} = \frac{C}{2\pi }\frac{1}{\sqrt{2h}} \quad \mathrm {and} \quad \left. \frac{\partial ^2 J_1}{\partial h^2} \right| _{j_2=0} = -\frac{C}{4\sqrt{2}\pi }\frac{1}{h^{3/2}}. \end{aligned}$$

Again differentiating under the integral sign gives

$$\begin{aligned} \left. \frac{\partial J_1}{\partial j_2} \right| _{j_2=0} = \frac{1}{2\pi } \oint \left. \frac{-j_2}{f^2\sqrt{2h-\frac{1}{f^2}j_2^2}} \, ds \right| _{j_2=0} = 0. \end{aligned}$$

Similarly, $\left. \frac{\partial ^2 J_1}{\partial h\partial j_2} \right| _{j_2=0} = 0$ as well. Finally,

$$\begin{aligned} \left. \frac{\partial ^2 J_1}{\partial j_2^2} \right| _{j_2=0} = \frac{1}{2\pi } \oint \frac{-1}{f^2\sqrt{2h}}\, ds . \end{aligned}$$

Using the chain rule we get

$$\begin{aligned} \frac{\partial H}{\partial j_1}&= \left( \frac{\partial J_1}{\partial h}\right) ^{-1} \\ \frac{\partial H}{\partial j_2}&= \frac{\partial J_1}{\partial j_2}\left( \frac{\partial J_1}{\partial h}\right) ^{-1} \\ \frac{\partial ^2 H}{\partial j_1^2}&= \left( \frac{\partial J_1}{\partial h}\right) ^{-2}\left[ \frac{\partial ^2 J_1}{\partial h^2}\frac{\partial J_1}{\partial h} + \frac{\partial ^2 J_1}{\partial h\partial j_2} \frac{\partial J_1}{\partial j_2} \right] \\ \frac{\partial ^2 H}{\partial j_1\partial j_2}&= \left( \frac{\partial J_1}{\partial h}\right) ^{-2}\left[ -\frac{\partial ^2 J_1}{\partial h^2} \frac{\partial J_1}{\partial j_2}\left( \frac{\partial J_1}{\partial h} \right) ^{-1} + \frac{\partial ^2 J_1}{\partial h\partial j_2} \right] \\ \frac{\partial ^2 H}{\partial j_2^2}&= \left( \frac{\partial J_1}{\partial h}\right) ^{-2}\left[ \frac{\partial ^2 J_1}{\partial j_2^2}\frac{\partial J_1}{\partial h} - \frac{\partial ^2 J_1}{\partial h\partial j_2} \frac{\partial J_1}{\partial j_2} \right] . \end{aligned}$$

Evaluating on the energy level $h=\frac{1}{2}$ we get

$$\begin{aligned} \begin{array}{rlc} \frac{\partial J_1}{\partial h} = \frac{C}{2\pi }, &{} \frac{\partial ^2 J_1}{\partial h^2} = -\frac{C}{2\pi }, &{} \frac{\partial ^2 H}{\partial j_1^2} =1, \\ \frac{\partial ^2 H}{\partial j_1\partial j_2} =0, &{} \mathrm {and} \, \, \, \frac{\partial ^2 H}{\partial j_2^2}= \frac{1}{C}\oint \frac{ds}{f^2} = \frac{M}{C} . &{} \end{array} \end{aligned}$$

Hence the linearized return map ${\mathcal {P}}_p$ is

$$\begin{aligned} \begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad M &{}\quad 1 &{}\quad 0 \\ C &{}\quad 0 &{}\quad 0 &{}\quad 1 \end{pmatrix} . \end{aligned}$$

Note that ${\mathcal {P}}_p$ is in normal form for the odd symplectic group. This normal form is different than the normal form for the equatorial case, since it is a unipotent matrix with two indecomposable blocks each having a nonzero modulus.