---
abstract: 'The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincaré sections to demonstrate the breakdown of regular motion.'
address: |
    Department of Mathematics, University of Portsmouth,\
    Portsmouth PO1 3HF, United Kingdom
author:
- 'Thomas J. Waters'
bibliography:
- 'geobib.bib'
title: Regular and irregular geodesics on spherical harmonic surfaces
---

geodesic ,integrability ,differential Galois theory ,spherical harmonics ,Kovacic algorithm ,Morales-Ramis

Introduction
============

The study of the qualitative behavior of geodesics goes back to the beginnings of differential geometry; in fact Clairault’s relation dates from the 1740’s. One of the many interesting features of the geodesic equations is their complexity: how can we solve such difficult nonlinear differential equations, and indeed what does it mean to ‘solve’ a differential equation at all? In the general development of Lagrangian/Hamiltonian dynamics the issue of integrability came to the fore, and the use of first integrals to solve a dynamical system via quadratures was developed. By integrability here we mean intebrability in the sense of Louiville: a Hamiltonian system with $n$ degrees of freedom is Louiville integrable if there exist $n$ independent first integrals of the motion in involution (vanishing pairwise Poisson brackets). This formalism in turn lead to the consequences of non-integrability, the onset of chaos and the celebrated KAM theory. For a particularly interesting introduction to this topic see Berry [@berry], McCauley [@mccauley] or José and Saletan [@jose].

The integrability of the geodesic equations on some surfaces, and therefore the regularity of the geodesic trajectories, has been well understood for many years. Clairault’s relation [@docarmo] provides a second integral for surfaces of revolution, and Jacobi [@kling] integrated the triaxial ellipsoid. These results have been extended to $n$-dimensional ellipsoids and to quadrics in general (see Audin [@audin]), where interestingly degenerate cases can prove more difficult to analyse (see Davison et al [@davison]). Recently, the integrability of the geodesic equations has been demonstrated for more abstract manifolds, defined in terms of Lie groups (see for example Thimm [@thimm]). Aside from being interesting in their own right, the geodesics of a manifold also have some useful applications. For example, Maupertius’s principle [@bols] relates the dynamics on an isoenergetic level of any mechanical system to the geodesics of a manifold, and the relativistic formulation of gravity dictates that massive particles move along the timelike geodesics of spacetime [@wald].

There are many powerful results regarding the non-integrability of the geodesic flow[^1] on manifolds of a certain class, for example a theorem of Kolokol’tsov [@kolok] says that two-dimensional compact surfaces of genus $g>1$ do not admit geodesic flows that are integrable with a second integral polynomial with respect to the momenta (see Bolsinov [@bols2] for a review and references). What’s more the integrable geodesic flows on surfaces with $g=0,1$ with second integrals either linear or quadratic in the momenta are completely described. Nonetheless, new examples of integrable geodesic flows with second integrals cubic in the momenta have recently been found (see for example Dullin and Matveev [@dullin]) and the possibility of non-polynomial, and in particular meromorphic second integrals, is open.

A very significant recent development with regards integrability has been the theorem of Morales Ruiz and Ramis [@mrthm1; @mrthm2]. This theorem provides a link between the question of whether a nonlinear Hamiltonian system is [*integrable*]{} (with meromorphic first integrals) with whether the variational equations of a non-trivial solution to the Hamiltonian system are [*solvable*]{}. This is a great advance as the variational equations are linear with variable coefficients, and the solvability of equations of this type may be addressed using differential Galois theory. This approach has proved very successful (we give here only some examples): in Morales Ruiz and Ramis [@mrthm2] the authors use this formalism to prove the non-integrability of certain problems from mechanics and celestial mechanics; in Morales Ruiz et al [@simon] the non integrability of Hill’s problem is shown; in Pujol et al [@sam] the non-integrability of the swinging Atwood’s machine is demonstrated; in Acosta-Humánez et al [@humanez] some problems in celestial mechanics are studied and in Bardin et al [@bardin] the generalized Jacobi problem is studied. In the majority of these papers, the solvability of the variational equations is determined using Kovacic’s algorithm, and this is the approach we will follow in this work, with a description of the algorithm and relevant references given in Section 4.

To the best of our knowledge this powerful Morales-Ramis formalism has not been applied to the study of geodesics on manifolds and their integrability properties, and the approach taken in this work is novel in this respect. In the following we will examine the properties of geodesics on surfaces defined in terms of the spherical harmonics, and we shall rigorously prove using the Morales-Ramis theorem the non-integrability of the geodesic equations on surfaces defined in terms of sectoral spherical harmonics. Thus we present a two-parameter family of surfaces which appear very simple and symmetrical and yet have complex and indeed chaotic geodesic curves on them; this is a novel addition as the surfaces for which non-integrability of the geodesic flow has been rigorously proved is small. Finally, we will also use Poincaré sections to display the breakdown of regular motion as well as to understand the various families of closed geodesics on the surfaces; again this approach is novel in the case of geodesic flows.

The layout of the paper shall be as follows: In Section 2 we review the formulation of the geodesic equations and describe the classes of surfaces which will be examined. In Section 3 we use Poincaré sections to exhibit the breakdown in regular motion suggesting non-integrability, as well as describe the main families of closed geodesics on the surface. Finally in Section 4 we will review some necessary theory and prove the main result: the non-integrability of the geodesic equations on sectoral harmonic surfaces with $n>1$.

Conclusions
===========

The application of Morales-Ramis theory and Poincaré sections, two techniques most often seen in the realm of celestial mechanics and mechanical systems, to study the integrability of geodesic flow presents many interesting possibilities. By choosing a specific class of surfaces to analyze we can rigorously prove the non-integrability of the geodesic equations. It would be of interest to extend the results of this work to examine further surfaces; for example the tesseral harmonics mentioned in Section 2 again have (meridianal) planes of symmetry, and the NVE will have coefficients that contain Legendre polynomials, and in general, it would be very informative if we could examine arbitrary surfaces with a plane of symmetry and try and arrive at some obstructions to integrability. We note that the presence of a plane of symmetry makes the decoupling of the NVE straightforward, however Morales Ruiz [@morruiz] outlines a procedure in the absence of an invariant plane. The completeness of the spherical harmonics means that surfaces of a very broad class can be decomposed into a set of spherical harmonics, and perhaps by analyzing these individually we can make some conclusions about their sum. Care must be taken however: the triaxial ellipsoid can be decomposed into a set of (infinite) spherical harmonics, each of which would likely have non-integrable geodesic flow, and yet their recombination gives a surface which [*is*]{} integrable. A further issue to consider is how the surface in question is described; the parametric description of the harmonic surfaces led easily to the calculation of the geodesic equations and in particular the separation of the normal variational equation. If the surface were defined in algebraic form, as is more suited to the double torus for example, this separation may be more problematic. Finally, this subject provides an interesting and fruitful combination of techniques from dynamical systems theory, differential geometry and differential Galois theory.

Acknowledgements
================

The author wishes to thank the School of Mathematics, National University of Ireland, Galway, where some of this work was carried out; Dr. Sergi Simon for useful conversations; and finally the referees for their important and constructive comments.

Meridianal sections
===================

The analysis above would carry through were we to use one of the meridianal planes instead of the equatorial plane, with two points to note:

1.  When drawing the Poincaré sections, the fact that $\phi=$constant is only a half plane complicates matters. Better to rotate the coordinate system (or surface) to make the meridian an equator. For this we need to rotate the surface about the $x$-axis, and for this we define a new set of coordinates $\vartheta,\varphi$ by $$\sin\theta=\sqrt{1-\sin^2\vartheta
    \sin^2\varphi},\quad \sin\phi=\frac{\cos\vartheta}{\sin\theta},
    \quad \cos\phi=\frac{\sin\vartheta \cos\varphi}{\sin\theta}.$$ Now to write the $n$th sectoral harmonic we expand $\cos(n\phi)$ and then make the replacement; for example, the 3rd sectoral harmonic changes as $$\begin{aligned}
     \hspace{-0.5cm} \sin^3\theta \cos(3\phi)=\sin^3\theta
    (\cos^3\phi-3\cos\phi\sin^2\phi)=\sin^3\vartheta\cos^3\varphi-3\sin\vartheta\cos\varphi\cos^2\vartheta.\end{aligned}$$ An example of a Poincaré section for this rotated surface is shown below; the same families of closed geodesics can be identified.

    ![Sample Poincaré section for the rotated $n=3$ sectoral harmonic surface with $\vep=0.1$. The elliptic nature of the equatorial geodesic (the fixed point at the centre of the plot) can be clearly seen.](merideps){width="80.00000%"}

2.  In deriving the variational equations we may linearize around a meridianal planar geodesic, to derive a normal variational equation of the form given in . In the case of an equatorial planar geodesic the denominators of the coefficients are at most quadratic in $z$, however in the case of a meridianal planar geodesic the denominators are polynomials of degree $2n+2$. While the singular points are still regular, the number, location and behaviour of the set of singular points will vary with $n$, complicating matters.

Coefficients of equation 
=========================

$$\begin{aligned}
  \delta_1&=\frac{2}{n(\vep^2-1)}, \\
  \delta_{2/3}&=\pm\frac{8+n^2\pm 2\vep(8+4n+3n^2)+e^2(8+8n+5n^2+8n^3+4n^4)}{16\vep(1\pm\vep)^2 n^2}, \\
  \delta_{4/5}&=-\frac{1}{2n(1-\vep^2)^2}\left( \vep^2(1-2n^2)-1\pm\big[n(1+\vep^2)\sqrt{1+\vep^2(n^2-1)}\,\big]
  \right) \\ &\qquad -\frac{1}{8}\left(\frac{4}{a_{4/5}-a_1}+\frac{1}{a_{4/5}-a_2}+\frac{1}{a_{4/5}-a_3}-\frac{1}{a_{4/5}-a_{5/4}}
  \right),\end{aligned}$$

where the $a_j$ are as defined in ,.

Bibliography
============

[^1]: Technically, the geodesic flow is defined on the cotangent bundle whose projection onto the manifold gives the geodesic curves, however we will use these two tems interchangeably.
