# Self-similarity in the Kepler-Heisenberg problem

Victor Dods<EMAIL_ADDRESS>and  Corey Shanbrom  California State
University, Sacramento, 6000 J St., Sacramento, CA 95819, USA
<EMAIL_ADDRESS>
###### Abstract.

The Kepler-Heisenberg problem is that of determining the motion of a planet
around a sun in the Heisenberg group, thought of as a three-dimensional sub-
Riemannian manifold. The sub-Riemannian Hamiltonian provides the kinetic
energy, and the gravitational potential is given by the fundamental solution
to the sub-Laplacian. The dynamics are at least partially integrable,
possessing two first integrals as well as a dilational momentum which is
conserved by orbits with zero energy. The system is known to admit closed
orbits of any rational rotation number, which all lie within the fundamental
zero-energy integrable subsystem. Here we demonstrate that, under mild
conditions, zero-energy orbits are self-similar. Consequently we find that
these zero-energy orbits stratify into three families: future collision, past
collision, and quasi-periodicity without collision. If a collision occurs, it
occurs in finite time.

###### 2010 Mathematics Subject Classification:

70H12, 70F05, 53C17, 65P10

This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1440140 while the first author was in residence at the
Mathematical Sciences Research Institute in Berkeley, California, during Fall
2018.

##  1\. Introduction

In geometric mechanics one usually constructs a dynamical system on the
cotangent bundle of a Riemannian manifold $(M,g)$ by taking a Hamiltonian of
the form $H=K+U$, where the kinetic energy $K$ is determined by the metric $g$
and the potential energy $U$ is chosen to represent a particular physical
system. In particular, the classical Kepler problem has been extensively
studied in spaces of constant curvature. These investigations date back to the
discovery of non-Euclidean geometry, when both Lobachevsky and Bolyai
independently posed the Kepler problem in hyperbolic 3-space. Serret,
Lipschitz, and Killing studied the Kepler problem on two and three dimensional
spheres. Modern researchers consider $n$ bodies in spaces of arbitrary
constant curvature $\kappa$. See [1] and the references therein for a thorough
history. In [6], we first posed the Kepler problem on the Heisenberg group in
the following manner.

Let $(\mathcal{H},D,\langle\cdot,\cdot\rangle)$ denote the sub-Riemannian
geometry of the _Heisenberg group_ :

  * •

$\mathcal{H}$ is diffeomorphic to $\mathbb{R}^{3}$ with usual global
coordinates $(x,y,z)$

  * •

$D$ is the plane field distribution spanned by the vector fields
$X:=\frac{\partial}{\partial x}-\frac{1}{2}y\frac{\partial}{\partial z}$ and
$Y:=\frac{\partial}{\partial y}+\frac{1}{2}x\frac{\partial}{\partial z}$

  * •

$\langle\cdot,\cdot\rangle$ is the inner product on $D$ which makes $X$ and
$Y$ orthonormal; that is, $ds^{2}=(dx^{2}+dy^{2})|_{D}$.

Curves tangent to the distribution have the property that their $z$-coordinate
always equals the area traced out by their projection to the $x,y$-plane. See
[5] for a detailed description of this geometry.

We define the Kepler problem on the Heisenberg group to be the dynamical
system on $T^{*}\mathcal{H}=(x,y,z,p_{x},p_{y},p_{z})$ with Hamiltonian

$H=\underbrace{\tfrac{1}{2}((p_{x}-\tfrac{1}{2}yp_{z})^{2}+(p_{y}+\tfrac{1}{2}xp_{z})^{2})}_{K}\underbrace{-\frac{1}{8\pi\sqrt{(x^{2}+y^{2})^{2}+{16}z^{2}}}}_{U}.$

The kinetic energy is $K=\frac{1}{2}(P_{X}^{2}+P_{Y}^{2})$, where
$P_{X}=p_{x}-\frac{1}{2}yp_{z}$ and $P_{Y}=p_{y}+\frac{1}{2}xp_{z}$ are the
dual momenta to the vector fields $X$ and $Y$; the flow of $K$ gives the
geodesics in $\mathcal{H}$. The potential energy $U$ is the fundamental
solution to the Heisenberg sub-Laplacian; see [4].

The resulting system is at least _partially integrable_ , in that both the
total energy $H$ and the angular momentum $p_{\theta}=xp_{y}-yp_{x}$ are
conserved. Moreover, the quantity

$J=xp_{x}+yp_{y}+2zp_{z}$ (1)

generates the Carnot group dilations in $T^{*}\mathcal{H}$, which are given by

$\delta_{\lambda}(x,y,z,p_{x},p_{y},p_{z})=(\lambda x,\lambda
y,\lambda^{2}z,\lambda^{-1}p_{x},\lambda^{-1}p_{y},\lambda^{-2}p_{z})$ (2)

for $\lambda>0$. It satisfies

$\dot{J}=2H$ (3)

and is thus conserved if $H=0$. Due to the three integrals $H$, $p_{\theta}$,
and $J$, the $H=0$ subsystem is integrable by the Liouville-Arnold theorem. It
is still unknown whether the entire system is integrable.

The zero-energy subsystem has been and continues to be our main focus. In [6]
we showed that periodic orbits must lie on the invariant hypersurface
$\\{H=0\\}$, and that the dynamics are integrable there. In [7] we showed that
periodic orbits do exist and in [3] that they enjoy a rich symmetry structure;
they realize every rational rotation number $j/k\in(0,1]$. Periodic orbits
must have $J=0$. Here we investigate orbits with $J\neq 0$ as well.

As in [3], much of this work was computer-assisted. Running extensive careful
numerical experiments led to many insights, including numerical verification
of the statement of Theorem 4.1 and the correct choice of dilation and time-
reparametrization factors (see Section 5), which made the formal proof
relatively simple. Our codebase can be found at [2].

In Section 2 we state the main result and discuss the mild assumptions
included and their potential removal. In Section 3 we introduce a new
coordinate system on $T^{*}\mathcal{H}$ specifically adapted for the Kepler-
Heisenberg problem, which separates the symmetries of the system. In Section 4
we state our result technically, derive some interesting consequences, and
provide some pictures. Finally, Section 5 contains the proof.

##  2\. Main Result and Discussion

We will call a Kepler-Heisenberg orbit $\gamma(t)=(x(t),y(t),z(t))$ an _A-
orbit_ if it satisfies:

$\displaystyle H=0$ (A1) $\displaystyle\gamma(t)\ \text{does not meet the
$z$-axis}$ (A2) $\displaystyle z(t)\ \text{has at least three zeros}.$ (A3)

The explanation for these conditions will constitute most of the the remainder
of this section. Our main result is the following, proved in Section 5.

######  Theorem 2.1.

Every Kepler-Heisenberg $A$-orbit is self-similar.

Figure 1. A sample self-similar orbit. Here and below, the left box shows the
projection of the orbit to the $x,y$-plane. The right box shows the
$z$-coordinate over time, which is equal to the area traced out by this
projection. The black dot represents the sun, the blue curve represents the
planet’s orbit, the purple segment a sampled fundamental domain, and the
orange segment the self-similar replication of the purple segment.

See Figure 1 for an example. The self-similarity here involves a dilation (see
(2)), a rotation, and a time-reparametrization. A more precise statement can
be found in Theorem 4.1. To the best of our knowledge these solutions
represent the first occurrence of this type of self-similarity within
solutions to ODEs of Hamiltonian type.

Note that due to the time-reparametrization, this self-similarity is not true
quasi-periodicity in the sense of “periodic modulo a transformation of space”.
However, zero-energy orbits are naturally stratified into three families,
depending on whether $J$ is negative, positive, or zero. Self-similarity for
orbits with $J=0$ does not require a time-reparametrization, so we find that
these orbits are genuinely quasi-periodic. See Corollary 4.2.

We now explain the origin of conditions (A1) – (A3) above, and provide some
conjectures regarding their necessity.

Condition (A1) is necessary as nearly everything we know about the Kepler-
Heisenberg system lies within the $H=0$ subsystem. As described in Section 1,
this subsystem is known to be integrable, and it contains all periodic orbits.

Condition (A2) is ostensibly a coordinate singularity. The backbone of our
proof of Theorem 2.1 is the correct choice of coordinate system, which
separates the dilation and rotation group actions. See Section 3. These charts
exclude the $z$-axis. However, this is not an artificial singularity: the
$z$-axis in the Heisenberg group is the conjugate and cut locus. Orbits with
initial configuration on the $z$-axis can behave strangely. For example, in
[6] we discovered orbits of the form $(0,0,c_{1},0,0,c_{2}t)$ for constants
$c_{1},c_{2}$. That is, a planet can sit at a stationary point on the $z$-axis
with linearly growing momentum. Note that such orbits satisfy $H<0$.

Numerical analysis of zero-energy orbits meeting the $z$-axis shows two
distinct types of behavior depending on the dilational momentum, with the
bifurcation occuring at the critical value $|J|=\frac{1}{2\sqrt{\pi}}$
appearing in Proposition 5.2. If $|J|<\frac{1}{2\sqrt{\pi}}$ then the orbit is
like a dilating figure-eight; as $J$ shrinks this family approaches the
rotationally asymmetric periodic figure-eight orbit shown in Figure 3 of [3].
These orbits are self-similar in the same way as the others considered in this
paper, and we believe our proof can be modified to include this case. However,
if $|J|>\frac{1}{2\sqrt{\pi}}$ then the orbit is like a dilating helical
Heisenberg geodesic. These are self-similar in a different way; in particular,
$z(t)$ is not oscillatory but monotonic. These planets escape the sun’s
gravitational pull in some sense. For $J>0$, for example, the necessary escape
momentum is determined by $J=xp_{x}+yp_{y}+2zp_{z}>\frac{1}{2\sqrt{\pi}}$; for
an orbit starting on the $z$-axis we simply need
$p_{z}>\frac{1}{4z\sqrt{\pi}}$. Work in this area is ongoing; the singular
geometry of the Heisenberg group along the $z$-axis means special care must be
taken. Our numerical investigation suggests the following.

######  Conjecture 2.2.

Any zero-energy orbit meeting the $z$-axis is self-similar.

Condition (A3) represents our failure so far to prove our suspicion that
$z(t)$ is oscillatory for any orbit satisfying (A1) and (A2). Here,
oscillatory is used in the sense of oscillation theory: that $z(t)$ has
infinitely many zeros. We have numerically verified that this is the case for
2,500 uniformly sampled orbits. We conducted a comprehensive survey of the
literature on Sturm comparison theorems and found no known version general
enough for the second-order, inhomogeneous, nonlinear ODE satisfied by $z(t)$,
so the question remains open.

######  Conjecture 2.3.

For any zero-energy orbit not meeting the $z$-axis, $z(t)$ is oscillatory.

There is a degenerate case: solutions for which $z$ is identically zero.
Proposition 3 of [6] shows that such orbits exist, but they are all
necessarily lines through the origin in the $x,y$-plane. These orbits are
trivially self-similar, so Theorem 2.1 holds. Note that if $z(t)=0$ on any
open interval, then the orbit must be one of these lines through the origin in
the plane by uniqueness of solutions to ODEs. So among orbits satisfying
condition (A3), we will restrict our attention to those with three discrete
zeros of $z$, and later in this paper we will refer to “consecutive” zeros
without reference to this degenerate case.

Note that we believe that Conjecture 2.3 also holds for orbits which do meet
the $z$-axis if $J<\frac{1}{2\sqrt{\pi}}$. More importantly, Conjectures 2.2
and 2.3 say that we suspect our conditions (A2) and (A3) can be removed from
Theorem 2.1. That is, we suspect the following.

######  Conjecture 2.4.

Every zero-energy orbit is self-similar.

##  3\. New Coordinates

Dilation by $\lambda>0$ and rotation about $z$-axis by $\varphi$ are important
transformations of configuration space $\mathcal{H}$. These naturally lift to
phase space $T^{*}\mathcal{H}$ as the maps $\delta_{\lambda}$ and
$\rho_{\varphi}$. The former is given in (2), and the latter by

$\rho_{\varphi}(x,y,z,p_{x},p_{y},p_{z})=\begin{bmatrix}\cos\varphi&-\sin\varphi&0&0&0&0\\\
\sin\varphi&\cos\varphi&0&0&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&\cos\varphi&-\sin\varphi&0\\\ 0&0&0&\sin\varphi&\cos\varphi&0\\\
0&0&0&0&0&1\\\ \end{bmatrix}\begin{bmatrix}x\\\ y\\\ z\\\ p_{x}\\\ p_{y}\\\
p_{z}\end{bmatrix}.$

Both maps are symplectomorphisms of $T^{*}\mathcal{H}$, and they satisfy
$H\circ\rho_{\varphi}=H$. and $H\circ\delta_{\lambda}=\lambda^{-2}H$. In the
$H=0$ case, both are symmetries, so we look for explicit coordinates in which
the corresponding moment maps are momentum coordinates. Since the rotation and
dilation actions commute, coordinates $(s,\theta,u)$ exist where dilations
correspond to a shift in $s$, rotations to a shift in $\theta$, and $u$ is
invariant under these actions. For rotations, usual polar coordinates
$(r,\theta)$ in the plane work, as $p_{\theta}$ is the desired conserved
quantity of angular momentum. However, finding coordinates with this property
for $J$ rather than $p_{\theta}$ is much more difficult.

We begin with position coordinates $(s,\theta,u)$ and insist that $p_{s}=J$.
Here $s$ and $u$ are unknowns to be solved for, and $\theta=\arg(x,y)$ is the
usual angle coordinate in the plane. For convenience we use the notation
$w=4z$ and $R=r^{2}$. Demanding canonical coordinates and $p_{s}=J$ yields the
following system of first-order, nonlinear PDEs:

$\displaystyle 2R\left(\frac{\partial s}{\partial R}\frac{\partial u}{\partial
w}-\frac{\partial u}{\partial R}\frac{\partial s}{\partial w}\right)$
$\displaystyle=\frac{\partial u}{\partial w}$ $\displaystyle
2w\left(\frac{\partial s}{\partial R}\frac{\partial u}{\partial
w}-\frac{\partial u}{\partial R}\frac{\partial s}{\partial w}\right)$
$\displaystyle=-\frac{\partial u}{\partial R}.$

Some algebra shows that $du$ must annihilate the radial vector field, so we
choose $u=\arg(R,w)$. This allows our system to be reduced to the first-order,
linear PDE

$R\frac{\partial s}{\partial R}+w\frac{\partial s}{\partial w}=\frac{1}{2},$

which can be solved for $s$ by the method of characteristics.

Thus the following coordinates have the desired properties:

$\displaystyle s$ $\displaystyle=\frac{1}{4}\log((x^{2}+y^{2})^{2}+16z^{2})$
$\displaystyle\theta$ $\displaystyle=\arg(x,y)$ $\displaystyle u$
$\displaystyle=\arg(x^{2}+y^{2},4z)$ $\displaystyle p_{s}$
$\displaystyle=xp_{x}+yp_{y}+2zp_{z}$ $\displaystyle p_{\theta}$
$\displaystyle=xp_{y}-yp_{x}$ $\displaystyle p_{u}$
$\displaystyle=\frac{1}{4}p_{z}(x^{2}+y^{2})-2z\frac{xp_{x}+yp_{y}}{x^{2}+y^{2}}.$

These are not global coordinates on $T^{*}\mathcal{H}$. Clearly $s$ is
singular at the origin, but this represents collision with the sun and is
actually a singularity for our system (technically, our phase space should
exclude the origin). However, $\theta$ is not defined for points on the
$z$-axis, so we must exclude this set from our chart. The $z$-axis is special
in the Heisenberg group: it is the cut and conjugate locus. See Section 2 for
further discussion. Note that $u$ represents an inclination angle, with $\arg$
here taking values in $(-\pi/2,\pi/2)$, and that $z=0$ if and only if $u=0$.

By construction, these are especially nice coordinates to use, and they
greatly simplify the expression of the rotation and dilation actions. One
easily checks by direct calculation that the following two results hold.

######  Proposition 3.1.

We have that $(s,\theta,u,p_{s},p_{\theta},p_{u})$ are canonical coordinates
on $T^{*}\mathcal{H}$ and that $p_{s}=J.$ Moreover, we have

$\displaystyle\rho_{\varphi}:$ $\displaystyle\theta\mapsto\theta+\varphi$
$\displaystyle\delta_{\lambda}:$ $\displaystyle s\mapsto s+\log\lambda$

with all other coordinates unaffected by these maps.

######  Proposition 3.2.

In $(s,\theta,u,p_{s},p_{\theta},p_{u})$ coordinates, our Hamiltonian takes
the form

$H(s,\theta,u,p_{s},p_{\theta},p_{u})=\exp\left(-2s\right)\left(\frac{1}{2}\left[\begin{array}[]{ccc}p_{s}&p_{\theta}&p_{u}\end{array}\right]\left[\begin{array}[]{ccc}\cos
u&\sin u&0\\\ \sin u&\sec u&2\cos u\\\ 0&2\cos u&4\cos
u\end{array}\right]\left[\begin{array}[]{c}p_{s}\\\ p_{\theta}\\\
p_{u}\end{array}\right]-\frac{1}{4\pi}\right).$

##  4\. Theorem, Consequences, Pictures

Recall that here we are only interested in orbits with $H=0$, for which the
dilational momentum $J$ is an integral of motion. In order to state the main
theorem, Theorem 4.1, it remains to compute the appropriate dilation,
rotation, and time-reparametrization factors. Details, along with the proof,
appear in Section 5. The coordinates $(s,\theta,u)$ and the maps
$\rho_{\varphi}$ and $\delta_{\lambda}$ are defined above in Section 3.
Consequently, Theorem 2.1 is a paraphrasing of the following.

######  Theorem 4.1.

Assume $c:I\to T^{*}\mathcal{H}$ is a solution to Hamilton’s equations on a
maximal domain $I$. Assume $H(c(t))=0$ for all $t$ and $c$ does not meet the
$z$-axis. Suppose $z(t)$ has consecutive zeros at $t_{0},t_{1},t_{2}$. Let

$\displaystyle\lambda$ $\displaystyle=\exp(s(t_{2})-s(t_{0}))$
$\displaystyle\varphi$ $\displaystyle=\theta(t_{2})-\theta(t_{0})$
$\displaystyle\xi(t)$
$\displaystyle=\frac{1}{2}\log_{\lambda}\left(1-(t-t_{0})\frac{1-\lambda^{2}}{t_{2}-t_{0}}\right)$
$\displaystyle\tau(t)$
$\displaystyle=t_{0}+(t_{2}-t_{0})\frac{1-\lambda^{2\xi(t)+2\lfloor\xi(t)\rfloor}}{1-\lambda^{2}}$

where we take $\xi(t)=\tau(t)=t$ if $\lambda=1$. Then for any time $t\in I$ we
have

$c(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ c\circ\tau(t).$

We refer to the interval $[t_{0},t_{2}]$ as a _fundamental time domain_. The
functions $\xi$ and $\tau$ are described more intuitively in Section 5.2; the
symbol $\lfloor\xi(t)\rfloor$ denotes the floor function. In Figure 2 we show
examples of self-similar orbits from each of the three families in Table 1. In
Figure 3 we show an example of numerical verification of the self-similarity
in phase space. This type of numerical investigation preceded and facilitated
both the statement and proof of Theorem 4.1.

Figure 2. Rows depict sample orbits with $\lambda<1$ , $\lambda>1$, and
$\lambda=1$, respectively. See Table 1. Colors and columns as in Figure 1.
Vertical lines show zeros of $z(t)$. Fundamental time domains lie between
three successive zeros. Successive domains scale by $\lambda^{2}$.

We now give some interesting consequences of Theorem 4.1; recall from Section
2 that A-orbits are those satisfying the hypotheses of the Theorem.

######  Corollary 4.2.

Every $A$-orbit with $J=0$ is quasi-periodic.

###### Proof.

If $\lambda=1$, then $\tau(t)=t$ and there is no time-reparametrization
involved, so the orbit is periodic modulo a transformation of space, namely
$\rho_{\varphi}\circ\delta_{\lambda}$. ∎

######  Corollary 4.3.

If an $A$-orbit collides, it does so in finite time.

###### Proof.

By definition of $\tau$, consecutive fundamental time domains scale by
$\lambda^{2}$. By Theorem 4.1, a zero-energy orbit suffers a collision in
future time if and only if $\lambda<1$. For such an orbit, the time to
collision is given by $\sum_{n=0}^{\infty}(t_{2}-t_{0})\lambda^{2n}$, a
convergent geometric series. Thus the collision time is

$t_{\text{col}}=t_{0}+\frac{t_{2}-t_{0}}{1-\lambda^{2}}.$ (4)

∎

######  Corollary 4.4.

$A$-orbits stratify into the families described in Table 1.

###### Proof.

This follows from Corollaries 4.2 and 4.3, since reversing the flow of time
inverts $\lambda$. ∎

Table 1. Summary of behavior of $A$-orbits Dilational momentum | Dilation factor | Behavior of orbit
---|---|---
$J<0$ | $\lambda<1$ | Future collision, unbounded past
$J>0$ | $\lambda>1$ | Past collision, unbounded future
$J=0$ | $\lambda=1$ | Periodic or quasi-periodic motion

Note that the results in Table 1 provide further evidence for our suspicion
that invariant submanifolds are diffeomorphic to
$\mathbb{T}^{2}\times\mathbb{R}^{+}$. The action of $J$ is non-compact, so we
do not quite have invariant 3-tori, but rather cylinders. When $J=0$, motion
is restricted to a 2-torus.

Figure 3. Example of numerical verification of self-similarity hypothesis in
phase space. A fundamental domain (purple) is replicated via self-similarity
(orange). It matches visually, but we also compute numerical error to verify.
Top row depicts positions $(x,y)$ and $(t,z)$ as in previous figures. Bottom
row depicts momenta $(p_{x},p_{y})$ and $(t,p_{z})$.

##  5\. Proof of Theorem

First we briefly sketch the proof. We extend an arbitrary $A$-orbit from a
fundamental domain by gluing scaled, rotated, and reparametrized copies
together. We show this extended curve is continuous and satisfies Hamilton’s
equations. Then by uniqueness of solutions to ODEs, the extended curve must be
the one and only orbit which extends our original orbit. Since the extended
curve is self-similar by construction, the unrestricted original orbit must be
as well. We begin by proving some technical results.

Throughout this section we use the notation of Theorem 4.1 and the coordinates
from Section 3. Assume $c:I\to T^{*}\mathcal{H}$ is a solution to Hamilton’s
equations on a maximal domain $I$. Assume $H(c(t))=0$ for all $t$. Suppose
$z(t)$ has consecutive zeros at $t_{0},t_{1},t_{2}$ and $c$ does not meet the
$z$-axis. Let

$\displaystyle\lambda$ $\displaystyle=\exp(s(t_{2})-s(t_{0}))$
$\displaystyle\varphi$ $\displaystyle=\theta(t_{2})-\theta(t_{0}).$

The orbit $c$ (and consequently the constants
$t_{0},t_{1},t_{2},J=p_{s},p_{\theta},\lambda,\varphi$) will be fixed
throughout; we will prove it is self-similar.

###  5.1. Preliminary results

######  Lemma 5.1.

For any $A$-orbit
$c(t)=(s(t),\theta(t),u(t),p_{s}(t),p_{\theta}(t),p_{u}(t))$, we have

$-\tan\left(u\right)p_{\theta}-\sqrt{2\tan^{2}\left(u\right)p_{\theta}^{2}+\frac{1}{4\pi\cos\left(u\right)}}\leq
p_{s}\leq-\tan\left(u\right)p_{\theta}+\sqrt{2\tan^{2}\left(u\right)p_{\theta}^{2}+\frac{1}{4\pi\cos\left(u\right)}}.$

###### Proof.

This is simply a calculation, rewriting the $H=0$ condition, which is
quadratic in the momenta coordinates. Using Proposition 3.2, we find that
$H=0$ is equivalent to

$p_{u}=\frac{-\cos\left(u\right)p_{\theta}\pm\sqrt{\frac{1}{4\pi}\cos\left(u\right)-\left(\cos\left(u\right)p_{s}+\sin\left(u\right)p_{\theta}\right)^{2}}}{2\cos\left(u\right)}.$

Setting the discriminant
$\Delta\left(p_{s}\right)=\frac{1}{4\pi}\cos\left(u\right)-\left(\cos\left(u\right)p_{s}+\sin\left(u\right)p_{\theta}\right)^{2}$
equal to zero gives

$p_{s}=-\tan\left(u\right)p_{\theta}\pm\sqrt{2\tan^{2}\left(u\right)p_{\theta}^{2}+\frac{1}{4\pi\cos\left(u\right)}}.$

Because $\Delta\left(p_{s}\right)$ is a negative-definite quadratic function
of $p_{s}$, it is the interior of these bounds that produces non-negative
values for $\Delta\left(p_{s}\right),$ and therefore

$-\tan\left(u\right)p_{\theta}-\sqrt{2\tan^{2}\left(u\right)p_{\theta}^{2}+\frac{1}{4\pi\cos\left(u\right)}}\leq
p_{s}\leq-\tan\left(u\right)p_{\theta}+\sqrt{2\tan^{2}\left(u\right)p_{\theta}^{2}+\frac{1}{4\pi\cos\left(u\right)}}.$

Note that condition (A2) implies $\cos(u)\neq 0$ and the expression under the
square root for these bounds is always positive since
$\cos\left(u\right)=\frac{R}{\sqrt{R^{2}+w^{2}}}$ and $R>0$.

∎

######  Proposition 5.2.

For any $A$-orbit we have $|J|\leq\frac{1}{2\sqrt{\pi}}$.

###### Proof.

Let $c(t)=(s(t),\theta(t),u(t),p_{s}(t),p_{\theta}(t),p_{u}(t))$ be such an
orbit. Then at some time $t_{0}$ we have $z(t_{0})=0$, so $u(t_{0})=0$. This
implies $\tan(u(t_{0}))=0$ and $\cos(u(t_{0}))=1$. Then by Lemma 5.1 we have
$-\sqrt{\frac{1}{4\pi}}\leq p_{s}(t_{0})\leq\sqrt{\frac{1}{4\pi}}$. But since
$H(c)=0$, we have that $p_{s}=J$ is constant in time, which completes the
proof. ∎

Proposition 5.2 is strong: it says there is a uniform bound on the dilational
momentum of all $A$-orbits. But this bound is also crucial in the proof of the
next result, which says that the end point of such an orbit on a fundamental
domain is a rotation and dilation of its start point. This Lemma is the
technical zenith of the proof of the main result; the rest will follow easily
and naturally.

######  Lemma 5.3.

We have $c(t_{2})=\rho_{\varphi}\circ\delta_{\lambda}\circ c(t_{0})$.

###### Proof.

The main idea here will be to show that

$p_{u}(t_{0})=p_{u}(t_{2});$ (5)

the other five coordinates are easy. First note that $u(t_{0})=u(t_{2})=0$;
that is, the endpoints here are characterized by lying in the $u=0$ plane
(which is also the $z=0$ plane). Therefore much of the subsequent analysis
will be focused on the behavior of the orbit when $u=0$. Imposing the two
constraints $u=H=0$ and applying Proposition 3.2 gives the following equation:

$\displaystyle 0$
$\displaystyle=\left[\begin{array}[]{ccc}p_{s}&p_{\theta}&p_{u}\end{array}\right]\left[\begin{array}[]{ccc}\cos
0&\sin 0&0\\\ \sin 0&\sec 0&2\cos 0\\\ 0&2\cos 0&4\cos
0\end{array}\right]\left[\begin{array}[]{c}p_{s}\\\ p_{\theta}\\\
p_{u}\end{array}\right]-\frac{1}{4\pi}$
$\displaystyle=4p_{u}^{2}+4p_{\theta}p_{u}+\left(p_{s}^{2}+p_{\theta}^{2}-\frac{1}{4\pi}\right).$

Thus,

$p_{u}|_{u=0}=-\frac{1}{2}p_{\theta}\pm\frac{1}{2}\sqrt{\frac{1}{4\pi}-p_{s}^{2}}.$

Now by Proposition 5.2, we have $\frac{1}{4\pi}-p_{s}^{2}\geq 0$, so this
expression is always defined. If $\frac{1}{4\pi}-p_{s}^{2}=0$, then
$p_{u}|_{u=0}=-\frac{1}{2}p_{\theta}$, which is constant, so clearly (5)
holds. Now assume $\frac{1}{4\pi}-p_{s}^{2}>0$. Let

$p_{u}^{\gamma}=-\frac{1}{2}p_{\theta}+\gamma\frac{1}{2}\sqrt{\frac{1}{4\pi}-p_{s}^{2}},$

where $\gamma\in\left\\{\pm 1\right\\}.$ From Hamilton’s equations,
$\frac{du}{dt}\rvert_{u=0}=\exp\left(-2s\right)\left(2p_{\theta}+4p_{u}|_{u=0}\right).$
But $p_{u}|_{u=0}$ is equal to $p_{u}^{\gamma}$ for some $\gamma$ that depends
on $t.$ Denote this time-dependent $\gamma$ value by $\gamma\left(t\right).$
Thus

$\left.\frac{du}{dt}\right|_{u=0}=\exp\left(-2s\right)\left(2p_{\theta}+4p_{u}^{\gamma\left(t\right)}\right)=2\gamma\left(t\right)\exp\left(-2s\right)\sqrt{\frac{1}{4\pi}-p_{s}^{2}}.$

Since $\frac{1}{4\pi}-p_{s}^{2}>0$, we have
$\text{sign}\left(\left.\frac{du}{dt}\right|_{u=0}\right)=\text{sign}\left(\gamma\left(t\right)\right),$
showing that the solution curve $c\left(t\right)$ intersects the plane $u=0$
transversally. Because $c$ is continuous, the sign of
$\left.\frac{du}{dt}\right|_{u=0}$ must alternate with each intersection. Thus
$\text{sign}\left(\gamma\left(t_{0}\right)\right)=-\text{sign}\left(\gamma\left(t_{1}\right)\right)=\text{sign}\left(\gamma\left(t_{2}\right)\right),$
and since $\gamma$ must take values in $\left\\{\pm 1\right\\},$ it follows
that $\gamma\left(t_{0}\right)=\gamma\left(t_{2}\right),$ and therefore that
$p_{u}\left(t_{0}\right)=p_{u}\left(t_{2}\right).$ Thus (5) holds in the case
that $\frac{1}{4\pi}-p_{s}^{2}>0$ as well.

Since $p_{s}$ and $p_{\theta}$ are conserved, we use (5), Proposition 3.1, and
the definitions of $\lambda$ and $\varphi$ to obtain

$\displaystyle\rho_{\varphi}\circ\delta_{\lambda}\circ c\left(t_{0}\right)$
$\displaystyle=\rho_{\varphi}\circ\delta_{\lambda}\left[\begin{array}[]{cc}s\left(t_{0}\right)&p_{s}\left(t_{0}\right)\\\
\theta\left(t_{0}\right)&p_{\theta}\left(t_{0}\right)\\\
u\left(t_{0}\right)&p_{u}\left(t_{0}\right)\end{array}\right]$
$\displaystyle=\left[\begin{array}[]{cc}s\left(t_{0}\right)+\log\exp\left(s\left(t_{2}\right)-s\left(t_{0}\right)\right)&p_{s}\left(t_{0}\right)\\\
\theta\left(t_{0}\right)+\left(\theta\left(t_{2}\right)-\theta\left(t_{0}\right)\right)&p_{\theta}\left(t_{0}\right)\\\
0&p_{u}\left(t_{0}\right)\end{array}\right]=\left[\begin{array}[]{cc}s\left(t_{2}\right)&p_{s}\left(t_{2}\right)\\\
\theta\left(t_{2}\right)&p_{\theta}\left(t_{2}\right)\\\
u\left(t_{2}\right)&p_{u}\left(t_{2}\right)\end{array}\right]=c\left(t_{2}\right).$

∎

###  5.2. Constructing a self-similar trajectory

We now aim to paste together self-similar copies of $c$ restricted to
$[t_{0},t_{2}]$. The main obstacle here is correctly reparametrizing time to
account for the dilation factor $\lambda$. If $\lambda=1$ then no such
reparametrization is necessary, so we temporarily assume $\lambda\neq 1$. To
this end, for any $\psi\in\mathbb{R}$, let

$\xi(t)=\frac{1}{2}\log_{\lambda}\left(1-(t-t_{0})\frac{1-\lambda^{2}}{t_{2}-t_{0}}\right)$
$\displaystyle\tau_{\psi}(t)$
$\displaystyle=t_{0}+(t_{2}-t_{0})\frac{1-\lambda^{2\xi(t)+2\psi}}{1-\lambda^{2}}$
$\displaystyle=t_{0}+(1-\lambda^{2\psi})\left(\frac{t_{2}-t_{0}}{1-\lambda^{2}}\right)+(t-t_{0})\lambda^{2\psi}.$

######  Lemma 5.4.

The function $\tau_{\psi}$ enjoys the following properties.

  1. (i)

The map $\psi\mapsto\tau_{\psi}$ is a homomorphism from the additive group of
real numbers to the group of invertible functions from $\mathbb{R}$ to
$\mathbb{R}$ under composition. That is, $\tau_{-\psi}=\tau_{\psi}^{-1}$ and
$\tau_{\psi}\circ\tau_{\eta}=\tau_{\psi+\eta}$ and
$\tau_{0}=\text{Id}_{\mathbb{R}}$.

  2. (ii)

For any $k\in\mathbb{Z}$, $\tau_{k}$ maps the fundamental domain
$[t_{0},t_{2}]$ to the appropriately scaled fundamental domain $k$ domains to
the right.

  3. (iii)

If $\psi\neq 0$ then $\tau_{\psi}$ has exactly one fixed point, which is the
collision time $t_{\text{col}}.$

###### Proof.

This is a straightforward and tedious calculation. ∎

We are finally in a position to extend $c$ from the domain $[t_{0},t_{2}]$ by
pasting together transformed copies. We define the curve $C$ inductively,
beginning with $C(t)=c(t)$ for all $t\in[t_{0},t_{2}]$. We then define $C$ on
the next fundamental domain as

$C(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ
c\circ\tau_{1}(t)\qquad\text{for all}\
t\in[t_{2},t_{2}+\lambda^{2}(t_{2}-t_{0})].$

We can continue to the next fundamental domain by letting

$C(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ
c\circ\tau_{2}(t)\qquad\text{for all}\
t\in[t_{2}+\lambda^{2}(t_{2}-t_{0}),t_{2}+\lambda^{2}(t_{2}-t_{0})+\lambda^{4}(t_{2}-t_{0})].$

In general, for $k\in\mathbb{Z}^{+}$, we have

$C(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ
c\circ\tau_{k}(t)\qquad\text{for all}\
t\in\left[t_{0}+(t_{2}-t_{0})\sum_{i=0}^{k-1}\lambda^{2i},t_{0}+(t_{2}-t_{0})\sum_{i=0}^{k}\lambda^{2i}\right].$
(6)

Similarly, we can define $C$ for past times, on the fundamental domain prior
to $[t_{0},t_{2}]$ as

$C(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ
c\circ\tau_{-1}(t)\qquad\text{for all}\
t\in[t_{0}-\lambda^{-2}(t_{2}-t_{0}),t_{0}],$

and generally for $k\in\mathbb{Z}^{-}$, by

$C(t)=\rho_{\varphi}\circ\delta_{\lambda}\circ
c\circ\tau_{k}(t)\qquad\text{for all}\
t\in\left[t_{2}-(t_{2}-t_{0})\sum_{i=0}^{k}\lambda^{2i},t_{2}-(t_{2}-t_{0})\sum_{i=0}^{k+1}\lambda^{2i}\right].$
(7)

Note that these expressions are still valid for the case of $\lambda=1$ if we
set $\tau_{\psi}(t)=t$ for all $\psi$. Thus we piece-wise construct the curve
$C$, whose domain is $(-\infty,t_{\text{col}})$ if $\lambda<1$,
$(t_{\text{col}},\infty)$ if $\lambda>1$, and all real numbers if $\lambda=1$.

The functions $\xi$, $\tau_{\psi}$ (above) and $\tau$ (Theorem 4.1) require
some explanation. Intuitively, $\xi$ dilates the original time $t$ so that
$z(\xi)$ is periodic with constant period $T=t_{2}-t_{0}$, rather than having
“periods” forming the geometric sequence $T,\lambda^{2}T,\lambda^{4}T,\dots$.
For an integer $k$, the function $\tau_{k}$ shifts the fundamental domain
$[t_{2},t_{0}]$ by $k$ fundamental domains to the right, as can be seen in
Equations (6) and (7) above. The function $\tau$ in Theorem 4.1 combines the
two in order to avoid the inductive construction of $C$ given in this section
and state the main theorem in a more concise and self-contained manner. More
precisely, the floor $\lfloor\xi(t)\rfloor$ computes the integer $k$ that
represents how many fundamental domains to the right of $[t_{0},t_{2}]$ we
must shift to contain the time $t$. That is,

$\tau=\tau_{\lfloor\xi\rfloor}.$

######  Proposition 5.5.

The curve $C$ is continuous.

###### Proof.

This follows from Proposition 5.3 and Lemma 5.4 (ii). By construction, the
pieces of $C$ agree at the endpoints (and $C$ is clearly continuous
elsewhere). ∎

######  Proposition 5.6.

The curve $C$ is smooth and satisfies Hamilton’s equations.

###### Proof.

This holds on $[t_{0},t_{2}]$ by assumption: $C=c$ there and $c$ was assumed
to be a solution. The maps $\rho_{\varphi}$ and $\delta_{\lambda}$ are
symmetries of the $H=0$ system ($\rho_{\varphi}$ is a symmetry of the full
system) and commute with the flow of the Hamiltonian vector field. That is,
they preserve solutions. This can also be seen in Proposition 3.1, as both
simply translate a single coordinate in the appropriate chart.

Now $\tau=\tau_{\lfloor\xi\rfloor}$ shifts the fundamental domain
$[t_{0},t_{2}]$ and dilates it by a factor of $\lambda^{2}$. The shift simply
gives us the correct domain for $C$. In [6], we showed that if a curve $c$
satisfies Hamilton’s equations, then so does
$\delta_{\lambda}(c(\lambda^{-2}t))$. In fact, we showed more generally that a
result like this holds for any homogeneous potential, and leads to a version
of Kepler’s third law. Here, we have that $H$ is homogeneous of degree two, as
$H\circ\delta_{\lambda}=\lambda^{-2}H$.

Thus each piece of $C$ is a solution. Since $C$ is continuous in phase space
we find that $C$ is a solution on all its domain. Since $H$ is smooth away
from the origin, it has a smooth Hamiltonian vector field, which has smooth
solutions, showing that $C$ is in fact smooth.

∎

###  5.3. Proof of Theorem 4.1

All the hard work is done, and Theorem 4.1 now follows from the Picard-
Lindelöf Theorem. Our Hamiltonian vector field is Lipschitz at the endpoints
of the fundamental domains since it is continuous on pre-compact neighborhoods
of these points. Thus $C$ is the unique solution to Hamilton’s equations and
extends $c$ to its maximal domain $I$ (to future or past collision).

### Acknowledgments

The authors are grateful to Richard Montgomery for many valuable
conversations. The first author would like to thank MSRI for the privilege of
taking part in the Fall 2018 Hamiltonian Systems semester, during which the
bulk of this project was completed.

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