Block regularisation of the logarithm central problem

TL;DR

This paper proves that the logarithm central force problem in two dimensions can be smoothly extended over the singularity at the origin through a process called block regularization, enabling better understanding of its dynamics.

Contribution

It introduces the concept of block regularization for the logarithm central force problem, showing the flow can be extended over the singularity after re-parametrization.

Findings

Flow can be continuously extended over the singularity

The problem is block regularizable after re-parametrization

Provides a new approach to singularity handling in gravitational potentials

Abstract

The logarithm function is the gravitational potential in $\mathbb{R}^2$. We prove that the logarithm central force problem is block regularizable, that is, the (incomplete) flow may be continuously extended over the singularity at the origin after an appropriate re-parametrization.

Equations

$$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x}),\mathbf{x}\in \mathbf{M}$$

$${\mathbf{b}}^{+}:=\{\mathbf{x}\in \mathbf{b}|\mathbf{\Theta }(\mathbf{x},(-\epsilon ,0))\cap \mathbf{B}=\varphi \text{for some}\epsilon >0\}$$

$${\mathbf{b}}^{-}:=\{\mathbf{x}\in \mathbf{b}|\mathbf{\Theta }(\mathbf{x},(0,\epsilon ))\cap \mathbf{B}=\varphi \text{for some}\epsilon >0\}$$

$$\displaystyle{\bf t}:=\left\{{\bf x}\in{\mathbf{b}}\,\Big{|}\,\,{\mathbf{F}}({\bf x})=\frac{d}{dt}\Big{|}_{t=0}\boldsymbol{\Theta}({\bf x},t)\,\,\text{is tangent to}\,{\mathbf{b}}\right\}$$

$$\frac{d}{dt}\Big{|}_{t=0}\left(\mathrm{I}(\boldsymbol{\Theta}({\bf x},t)\right))=0$$

$$\frac{d^{2}}{dt^{2}}\Bigg{|}_{t=0}\left(\mathrm{I}(\boldsymbol{\Theta}({\bf x},t)\right))>0.$$

$${\mathbf{a}}^{+}:=\{\mathbf{x}\in {\mathbf{b}}^{+}|\mathbf{\Theta }(\mathbf{x},[0,\infty ))\subset \mathbf{B}\}$$

$${\mathbf{a}}^{-}:=\{\mathbf{x}\in {\mathbf{b}}^{-}|\mathbf{\Theta }(\mathbf{x},(-\infty ,0])\subset \mathbf{B}\}.$$

$$t_{\text{exit}}(\mathbf{x}):=\text{inf}\{t>0|\mathbf{\Theta }(\mathbf{x},t)\notin \mathbf{B}\}$$

$$\mathbf{\Psi }(\mathbf{x})=\mathbf{\Theta }(\mathbf{x},t_{\text{exit}}(\mathbf{x})).$$

$${\mathbf{a}}^{+}:=\{\mathbf{x}\in {\mathbf{b}}^{+}|\mathbf{\Theta }\left(\mathbf{x},t\right)\subset \mathbf{B}\text{for all t for which Θ(x,t) is defined}\}$$

$${\mathbf{a}}^{-}:=\{\mathbf{x}\in {\mathbf{b}}^{-}|\mathbf{\Theta }\left(\mathbf{x},t\right)\subset \mathbf{B}\text{for all t for which Θ(x,t) is defined}\}.$$

$$H(\mathbf{q},\mathbf{p})=\frac{1}{2}{\mathbf{p}}^2+\ln (|\mathbf{q}|)$$

$$\dot{\mathbf{q}}=\mathbf{p},\dot{\mathbf{p}}=-\frac{\mathbf{q}}{|\mathbf{q}{|}^2}.$$

$$H(r,\theta ,p_r,p_{\theta })=\frac{1}{2}\left(p_r^2+\frac{p_{\theta }^2}{r^2}\right)+\ln (r)$$

$$\dot{r}=p_r,{\dot{p}}_r=\frac{p_{\theta }^2}{r^3}-\frac{1}{r},$$

$$\dot{\theta }=\frac{p_{\theta }}{r^2},{\dot{p}}_{\theta }=0.$$

$$H_{\text{red}}(r,p_r)=\frac{1}{2}{p}_r^2+\frac{c^2}{2r^2}+\ln (r)$$

$$V_{\text{red}}(r):=\frac{c^2}{2r^2}+\ln (r)\le h$$

$$\frac{dr}{dt}=p_r$$

$$\frac{d{p}_r}{dt}=\frac{c^2}{r^3}-\frac{1}{r}$$

$$\mathbf{q}=r{e}^{-\frac{1}{r^2}}\mathbf{s},\mathbf{p}=\frac{1}{r}\mathbf{u}$$

$$\frac{1}{2}|\mathbf{u}{|}^2+r^2\ln r-1=h{r}^2.$$

$$h{r}^2-r^2\ln r+1\ge 0.$$

$$\mathbf{s}=(\cos \varphi ,\sin \varphi ),\mathbf{u}=e^w(\cos \psi ,\sin \psi ).$$

$$d\tau =-\frac{1}{r^2}\exp \left(\frac{1}{r^2}-w\right)dt$$

$$\dot{r}=-\frac{r^3}{r^2+2}{e}^{2w}\cos \phi$$

$$\dot{\phi }=(e^{2w}-r^2)\sin \phi ,$$

$$\dot{\psi }=r^2\sin \phi ,$$

$$\dot{w}=r^2\left[1-\frac{e^{2w}}{r^2+2}\right]\cos \phi \text{with}(r,\phi ,\psi ,w)\in (0,\infty )\times S^1\times S^1\times \mathbb{R}$$

$$f(r)=\{\begin{array}{l}{r}^2\ln r,r>0,\\ 0,r=0,\end{array}$$

$$\frac{1}{2}{e}^{2w}=h{r}^2-f(r)+1.$$

$$\exp \left(w-\frac{1}{r^2}\right)\sin \phi =const.=:c$$

$$g(r)=\{\begin{array}{l}\exp \left(-\frac{1}{r^2}\right),r>0\\ 0,r=0.\end{array}$$

$$e^{w}g(r)\sin \phi =const.=:c.$$

Full text

Keywords: logarithm central force problem, singularity blow-up, block regularization

1 Introduction

During by the space race of the second half of the last century research in celestial mechanics experienced an invigorating period. A multitude of problems had to be solved at the theoretical and practical level. Amongst these, the singularity at collision in the equations of motion lead to extenuatory numerical approximations, as a smaller and smaller integration time interval was required to maintaining the model accuracy. [StSc71]. In this context, regularizing procedures became necessary, that is methods of transforming the equations, so that the “new” flow, identical up to a parametrization to the initial one, could be extended at least continuously over the singularity.

The elegant work of Levi-Civita, completed at the beginning of the 20th century [LC13], was brought in the limelight in [StSc71]: on a negative fixed level of energy, the planar Kepler problem may be transformed into to a harmonic oscillator with frequency depending on (the negative) energy. The method of Levi-Civita was extended to include homogeneous potentials of the form $\displaystyle{V(r)=-1/r^{2-2/n}},$ $n\geq 2,$ $n\in\mathbb{N}$ ($r$ being the distance from the particle to the centre) by McGehee [McGe81]. This was achieved while comparing two different regularisation methods: the “branch regularization” of Sundman [Su07] and “block regularisation”, or “regularisation by surgery” designed by Conley & Easton [CoEa71]. The first extends the double collisions as convergent power series in time in the complex plane. The second uses orbits that pass nearby the origin to extend, at least continuously, the flow past collision. The regularisation of the Kepler problem and connections with various mathematical physics fields was, and continues to be, the subject of a multitude of papers: [Mo70, Mil83, HedL12, GBM08, San09, LZ15, vdM21, ChHs22], to name a few.

The gravitational potential $(-1/r)$ is the fundamental solution of the Laplace equation in three dimensions. In two dimensions, the natural gravitational potential, understood as a solution of the same equation, is $V(r)=\ln r$. Logarithm potentials are used in astrophysics, in attempts to construct self-consistent models of galaxies; see for example [BiTr87, MESc89], and more recently, [BBP07, VWD12]. When compared to its Newtonian counterpart, the logarithm central force problem is also integrable, but all its trajectories are bounded (there is no escape to infinity) and not necessarily closed. As an attractive law, the logarithm law pull is weaker than in any law of the form $-1/r^{\alpha},$ $\alpha>0$ at close range, but stronger at large range. Not much is known about “logarithm” $n$-body problems, except for astrophysicists’ numerical simulations [BiTr87, MESc89, BBP07, VWD12]. In the context of regularisation via smoothing, in [CaTe11] it is proven that in the logarithm central force problem solutions ending/emerging in/from collision may be replaced with transmission trajectories. A study concerning the anisotropic two body problem is given in [StFo03]. No studies on the problem for $n\geq 3$ seem to exist.

Loosely speaking, an incomplete flow is block regularizable if solutions that asymptotically end in the singularity set (in our case the collision set) are in a bijective correspondence to solutions asymptotically leaving the singularity set. First the singularity set is blown up into an invariant manifold pasted into the phase space, so that the transformed flow is complete. Then one defines the map across the block that associates solutions that enter a neighbourhood of the invariant manifold replacing the singularity to solutions that exit the same neighbourhood. If this map can be at least continuously extended to solutions that asymptotically tend to/leave the invariant set, then the transformed flow is called to be trivializable, and the initial incomplete flow is block regularizable.

In this paper we prove that the logarithm central force problem is block regularizable. As in McGehee [McGe81], the blown up singularity set, called the collision manifold, is an invariant torus pasted into the phase space for all levels of energy. Since the logarithm function does not allow the use of the analytic transformations as in [McGe81], for the blown-up procedure we use ${\cal C}^{1}$ transformations similar to those in [StFo03]. However, the loss of smoothness does not impede our further analysis. The main result is proven of the bases of two facts: first, collisions are possible only for zero angular momenta and second, on the torus collision manifold, the flow takes a very simple form.

The work is organised as follows: in Section 2 we briefly introduce Conley and Easton’s [CoEa71] theory on trivializable isolating blocks for complete vector fields. In Section 3 we define block regularisation for singular of vector fields. In the next sections we discuss the logarithm central force problem, regularise the equations and the integrals of motion, and prove our main result.

2 Invariant manifolds, isolating blocks and trivializable flows

Consider a time-reversible system

[TABLE]

where $\mathbf{M}$ is a smooth manifold in $\mathbb{R}^{n}$, and ${\mathbf{F}}$ is ${\cal{C}}^{\infty}$ on its domain.

We assume that the flow of (1) is complete and denote it by $\boldsymbol{\Theta}({\bf x},t)$, ${\bf x}\in\mathbf{M}$. A compact flow-invariant set $\mathbf{N}\subset\mathbf{M}$ is called isolated if there is an open set $\mathbf{U}\supset\mathbf{N}$ such that if $\boldsymbol{\Theta}({\bf x},\mathbb{R})\subset\mathbf{U}$ then ${\bf x}\in\mathbf{N}.$ The set $\mathbf{U}$ is called an isolating neighbourhood for $\mathbf{N}.$

Consider $\mathbf{B}$ a compact subset of $\mathbf{M}$ with non-empty interior and assume that ${\mathbf{b}}:=\partial\mathbf{B}$ is a smooth submanifold of $\mathbf{M}$. Define (see Figure 1)

[TABLE]

Definition 2.1

If ${\bf t}={\mathbf{b}}^{+}\cap{\mathbf{b}}^{-}$ then $\mathbf{B}$ is called to be an isolating block.

Definition 2.2

An isolating block $\mathbf{B}$ isolates $\mathbf{N}$ if int($\mathbf{B}$) is an isolating neighbourhood for $\mathbf{N}$.

By a theorem of Conley and Easton [CoEa71], any isolated invariant set accepts an isolating block, and vice-versa, any isolating block contains an isolating block (possible empty).

A natural way to construct isolated blocks is by using a Lyapunov-type function.

Theorem 2.3** (Wilson and Yorke [WilYor73])**

Let $\mathrm{I}:\mathbf{M}\to[0,\infty)$ be a smooth function and $\delta_{0}>0.$ Suppose $D\mathrm{I}({\bf x})\neq 0$ for all ${\bf x}$ such that $0<\mathrm{I}({\bf x})\leq\delta_{0}$, and whenever $0<\mathrm{I}({\bf x})\leq\delta_{0}\,$ and

[TABLE]

we have,

[TABLE]

Then $\mathbf{N}:=\mathrm{I}^{-1}(0)$ is an isolated invariant set and $\mathrm{I}^{-1}\left([0,\delta]\right)$ is an isolating block for $\mathbf{N}$ for each $\delta\in(0,\delta_{0}].$

The subsets of ${\mathbf{b}}$ asymptotic to $\mathbf{N}$ are

[TABLE]

The map across the block $\boldsymbol{\Psi}:{\mathbf{b}}^{+}\setminus{\bf a}^{+}\to{\mathbf{b}}^{-}$ is constructed by assigning to each ${\bf x}\in{\mathbf{b}}^{+}\setminus{\bf a}^{+}$ the point along the flow $\boldsymbol{\Theta}({\bf x},t_{\text{exit}}({\bf x}))$, where

[TABLE]

that is

[TABLE]

In this context we have

Theorem 2.4

[Conley and Easton [CoEa71]] If $\mathbf{B}$ is an isolating block, then the map across the block $\boldsymbol{\Psi}:{\mathbf{b}}^{+}\setminus{\bf a}^{+}\to{\mathbf{b}}^{-}\setminus{\bf a}^{-}$ is a diffeomorphism.

Definition 2.5

The isolating block $\mathbf{B}$ is trivialisable if the map across the block $\boldsymbol{\Psi}$ extends to a diffeomorphism from ${\mathbf{b}}^{+}$ to ${\mathbf{b}}^{-}.$

3 Block regularisation of a vector field singularity

Consider a vector field (1) undefined on a compact set $\hat{\mathbf{N}}\subset\mathbf{M}$ so that trajectories approach $\hat{\mathbf{N}}$ in finite time. The flow is not complete and for each ${\bf x}\in\mathbf{M}$ we denote its orbit ${\cal O}({\bf x}):=\{{\boldsymbol{\Theta}}({\bf x},t)\,|\,\boldsymbol{\Theta}({\bf x},t)\,\,\text{is well-defined}\}.$ The definitions of $\mathbf{B}$, ${\mathbf{b}}$, ${\mathbf{b}}^{\pm}$ from the previous section remain the same, with the understanding that the orbit of points in ${\mathbf{b}}$ may exist for a finite time only. The same applies for the subsets ${\bf a}^{\pm},$ that is

[TABLE]

The map ${\boldsymbol{\Psi}}$ is defined in the same manner and the definition of $\mathbf{B}$ being trivialisable remains the same. Finally,

Definition 3.1

The vector field (1) is block regularisable over the singularity set $\hat{\mathbf{N}}$, or the singularity set $\hat{\mathbf{N}}$ is block regularisable, if there is a trivialisable block that isolates $\hat{\mathbf{N}}$.

Assume that there exists a (at least) ${\cal C}^{2}$ transformation of the variables and a time-reparametrisation, so that when applied to (1) the set $\hat{\mathbf{N}}$ is blown up (or transformed) into a set $\mathbf{N}$ pasted into the phase space that is approached by the (re-parametrized) flow asymptotically and for which $\mathbf{N}$ is a compact invariant manifold. We say then that the equations of motion are regularised. Notice that the flow on invariant set $\mathbf{N}$ is fictitious, but due to continuity with respect to initial data, it provides information on the behaviour of the real flow near $\hat{\mathbf{N}}$.

If the blow-out procedure is applicable, then the definition 3.1 above is equivalent to

Definition 3.2

The vector field (1) is block regularisable over the singularity set $\hat{\mathbf{N}}$ if there is a trivialisable block that isolates the blown-up invariant manifold $\mathbf{N}$.

4 The logarithm central force problem. Regularisation of the equations

Consider the motion of a unit mass particle described by the Hamiltonian $H:\mathbb{R}^{2}\setminus\{{\bf 0}\}\times\mathbb{R}^{2}\to\mathbb{R}$

[TABLE]

The equations of motion are

[TABLE]

In polar coordinates, the Hamiltonian reads

[TABLE]

and the equations of motion are

[TABLE]

By physics laws, or by direct observation, one may deduce that along any solution $\left(r(t),\theta(t),p_{r}(t),p_{\theta}(t)\right)$ the total energy $H(r(t),\theta(t),p_{r}(t),p_{\theta}(t))$ and the angular momentum $p_{\theta}(t)$ are conserved and thus can be considered as parameter. Denoting $p_{\theta}=:c$, we reduce the dynamics to a one degree of freedom system given by the reduced (or amended) Hamiltonian

[TABLE]

Since along any motion we have $H_{\text{red}}(r(t),p_{r}(t))=const.=:h$, the Hill regions of motion are given by inequality

[TABLE]

It follows that collision are possible only for zero angular momentum (i.e. for $c=0$), and that for any $c\neq 0$ the motion is possible for $h\geq h_{min}(c)=1/2+\ln(|c|)$. For all levels of energy the motion is always bounded, with the bounds depending on the parameters $(h,c)$, that is $r(t)\in[r_{min}(h,c),r_{max}(h,c)]\,.$

The equations of motion associated to (15) are

[TABLE]

By the standard ODE theory, for any initial conditions ${\bf x}(0)$ there is a unique solution defined on a maximal interval of existence $(t^{-},t^{+})$, with $-\infty\leq t^{-}<0<t^{+}\leq\infty$. Given that the system is time-reversible, $|t^{-}|=t^{+}$, and so the solution exists for $t\in(-t^{+},t^{+}).$ If $t^{+}<\infty$ , then the solution experience a singularity at $t^{*}=t^{+}$. In our case, a proof similar to that in [McGe81] shows that if solution experience a singularity at $t^{*}$, then $\displaystyle{\lim\limits_{t\to t^{*}}r(t)\to 0}$.

We proceed now to regularize the equations. First, in (11) we apply the change the variables $({\mathbf{q}},{\mathbf{p}})\to({\mathbf{s}},{\bf u})$:

[TABLE]

where $(r,{\mathbf{s}},{\bf u})\in(0,\infty)\times S^{1}\times\mathbb{R}^{2}$. In these coordinates the conservation of energy reads

[TABLE]

The system (19) is analytic on its domain and that the regions of motion are constrained by the energy relation (20). Notice that the kinetic term in (20) is positive, we have that for a fixed level of energy h

[TABLE]

It follows that for any fixed level of energy $h\in\mathbb{R}$, the motion exists and the trajectory is bounded, i.e., $0\leq r\leq r_{\text{Max}}(h).$ We now express the coordinates $({\mathbf{s}},{\bf u})$ in terms of $(w,\phi,\psi)\in\mathbb{R}\times S^{1}\times S^{1}$:

[TABLE]

Defining further $\varphi:=\phi-\psi$ and introducing the time-reparametrisation

[TABLE]

the equations of motion become

[TABLE]

(where, by abuse of notation, we denote the differentiation with respect to $\tau$ by “dot” as well). Notice that the vector field has a smooth extension to $r=0$. To extend the energy relation $\eqref{eq:energy}$, we define

[TABLE]

and observe that $f$ is of ${\cal C}^{1}$ class. This yields,

[TABLE]

Notice that the extended flow (23)-(26) is complete; we denote it by $\theta_{t}$. Further, notice that $\psi$ is determined by $r$ and $\varphi$ and, for $r$ small but positive, since $e^{2w}-r^{2}=\left[(2h-1-\ln r)r^{2}+1\right]\geq 0$

• •

if $\varphi\in(-\pi/2,\pi/2)$ then $r$ is decreasing, and furthermore, if $\varphi\in(-\pi/2,0)$ then $\varphi$ is decreasing, whereas if $\varphi\in(0,\pi/2)$ then $\varphi$ is increasing

• •

if $\varphi\in(\pi/2,3\pi/2)$ then $r$ is decreasing, and furthermore, if $\varphi\in(\pi/2,\pi)$ then $\varphi$ is increasing, whereas if $\varphi\in(\pi,3\pi/2)$ then $\varphi$ is decreasing.

In $(r,\varphi,\psi,w)$ coordinates, the angular momentum integral is

[TABLE]

and it is undefined at $r=0.$ By introducing

[TABLE]

we extend (28) to a well-defined and differentiable function, including points with $r=0$. The extended angular momentum relation reads:

[TABLE]

5 Block regularisation of the logarithm problem

As mentioned above, the flow of (23)-(26) is complete. For this flow, the collision set $\{r=0\}$ was blown up to the compact set

[TABLE]

which is an invariant manifold, called the collision manifold, pasted into the phase space for all levels of energy. The flow on $\mathbf{N}$ is fictitious and has no physical meaning, however, by the continuity with respect to initial data, one may extract information on the real flow behaviour near collision. On $\mathbf{N}$ equation of motion are

[TABLE]

and so, on $\mathbf{N}$ the flow is trivial, given by trajectories with $\psi(\tau)=const.=\psi_{0}.$ There are two circles of equilibria at $(\varphi,\psi)=(0,\psi_{0})$ and $(\varphi,\psi)=(\pi,\psi_{0})$ for all $\psi_{0}\in S^{1}$. The next lemma shows that the collision manifold is approached asymptotically in time.

Lemma 5.1

Let $h$ be fixed and let $(r,\varphi,\psi,w)$ be a solution of (23)-(26). Assume that $r\to 0$ as $t\to t^{*}\pm.$ Then $\tau(t)\to\pm\infty$.

Proof. We prove that $t\to t^{*}-$ implies $\tau(t)\to-\infty;$ the other case is analogous. The re-parametrisation (22) yields

[TABLE]

Thus $\tau(t)$ is defined for all $t<t^{-}$ and is decreasing. Thus $\lim\limits_{t\to t^{*}-}\tau(t)=\tau^{*}\geq-\infty.$ Then $\lim\limits_{\tau\to\tau^{*}}r(\tau)=0.$ But equations (23)-(26) are well-defined and with a smooth vector field. Since $\mathbf{N}$ (where $r=0$) is a compact invariant set, trajectories cannot approach it in finite time. Thus $\tau^{*}=-\infty.$

$\square$

Define the sets

[TABLE]

and denote by $\boldsymbol{\omega}(P)$ the omega limit set of a point $P=(r,\varphi,\psi,w)$ under the flow of (23)-(25) on a fixed level of $h$.

Lemma 5.2

For $h$ fixed, consider a solution $(r(\tau),\varphi(\tau),\psi(\tau),w(\tau))$ of the (23)-(26), with some initial condition $P_{0}:=(r(0),\varphi(0),\psi(0),w(0))$. Then $r(\tau)\to 0$ as $\tau\to\infty$ if and only if $\boldsymbol{\omega}(P_{0})={\bf S}^{+}$, and $r(\tau)\to 0$ as $\tau\to-\infty$ if and only if $\boldsymbol{\alpha}(P_{0})={\bf S}^{-}.$

Proof: We will only prove that $r(\tau)\to 0$ as $\tau\to\infty$ if and only if $\boldsymbol{\omega}(P_{0})={\bf S}^{+}$ since the other one is similar. Let $\theta_{\tau}(P_{0})$ denote the trajectory starting at $P_{0}$. Sufficiency follows from that fact that if $\boldsymbol{\omega}(P_{0})={\bf S}^{+}$ then, by definition, $\theta_{\tau}(P_{0})\rightarrow y,y\in\bf S^{+}$. But this implies that $r_{\tau}\rightarrow 0$.

To prove the converse assume that $r(\tau)\rightarrow 0$. Then $\bf S^{+}\subseteq\boldsymbol{\omega}(P_{0})$, so we need to prove the reverse inclusion. From the continuity of the angular momentum equation, if $r\rightarrow 0$, we conclude that $c=0$. Thus, any trajectory going to $\mathbf{N}$ must have angular momentum 0. For such a trajectory, we must have $\sin\varphi=0$ from the angular momentum equation. By continuity, this means that $\varphi(0)\in\{0,\pi\}$ and hence, $\varphi$ remains constant for all $\tau\in[0,\infty)$. If $\varphi(\tau)=\pi$ for all $\tau\geq 0$ then in a sufficiently small neighborhood of N, $r$ increases and hence $r(\tau)\nrightarrow 0$ as $\tau\rightarrow\infty$. We conclude that $\varphi(0)=\varphi=0$. Since, by assumption, $r(\tau)\rightarrow 0$, we conclude that $\boldsymbol{\omega}(P_{0})\subseteq\bf S^{+}$.

$\square$

Lemma 5.3

Let $(r,\varphi,\psi,w)$ a solution of (23)-(26) such that $r\to 0$ as $\tau\to\pm\infty.$ Then, as $\tau\to\pm\infty$,

[TABLE]

Proof. Following a similar analysis as in the previous proof, we conclude that $r\to 0$ as $\tau\to\pm\infty$ implies $c\equiv 0$, and consequently $\sin\varphi(\tau)\equiv 0$. The remaining conclusions follow by using equation (25) with $\sin\varphi=0$ and the energy equation.

$\square$

For a given $h$, using the conservation of energy equation, 27, we define

[TABLE]

It follows that the flow of (23) - (26) is a map $\mathbf{\Sigma}_{h}\times\mathbb{R}\rightarrow\mathbb{R}$. Define $\mathrm{I}:\mathbf{\Sigma_{h}}\rightarrow[0,\infty)$ by $\mathrm{I}(r,\varphi,\psi,w)=r$. The tangent space to a point $(r,\varphi,\psi,w)\in\mathbf{\Sigma}_{h}$ is

[TABLE]

Note that, for a fixed $h$, $r$ can be chosen small enough and positive so that $f^{\prime}(r)-2hr=(2\ln r+1-2h)r\neq 0$. This means that, there exists $\delta>0$ such that if $0<r<\delta$ then $D\mathrm{I}(r,\varphi,\psi,w)=\dot{r}\neq 0$. Next, assume that

[TABLE]

and $\mathrm{I}(r(0),\varphi(0),\psi(0),u(0))=r(0)=:r_{0}>0$. From (23), for $0<r_{0}<\delta$ we must have $\cos\varphi(0)=0$. Then

[TABLE]

assuming $\delta$ is sufficiently small.

Let

[TABLE]

The following proposition is a consequence of the Wilson and Yorke Lemma 2.3 and our prior discussion:

Proposition 5.4

$\mathbf{B}$* is an isolating block for the isolated invariant set *N.

Denote by $w_{\delta}$ the solution of $\eqref{EEO}$ with $r=\delta$. We let

[TABLE]

and we define, (see Figure 2)

[TABLE]

Denoting by $\boldsymbol{\Phi}_{t}(r,\varphi,\psi,w)$ the flow of (23)-(26), notice that the definitions above are in agreement with the definitions (2) - (6). Indeed, if $\cos\varphi(0)\leq 0$ and $r(0)=\delta$, then $r$ increases in $(0,\epsilon)$ for sufficiently small $\epsilon>0$. In that case, $\boldsymbol{\Phi}_{(0,\epsilon)}(\delta,\varphi,\psi,w_{\delta})\cap\bf B=\emptyset$ and therefore,

[TABLE]

where equality holds since $\cos\varphi(0)>0$ implies $r$ decreases and hence for every $\epsilon>0$, $\Phi_{(0,\epsilon)}(\delta,\varphi,\psi,w_{\delta})\cap\bf B\neq\emptyset$. In a similar vein,

[TABLE]

and ${\bf t}=\{(r,\varphi,\psi,w)\,|\,\dot{\boldsymbol{\Phi}}_{0}(r,\varphi,\psi,w)\text{\> is tangent to \>}{\mathbf{b}}\}$ by definition of an isolating block.

Next, if $\varphi=0$ along the flow then from (23)-(26), it follows by substituting $\varphi=0$ that $\dot{r}<0$. So, if $r(0)=\delta$, then the flow stays inside the block. Hence, we have, ${\bf a}^{+}\subseteq\{(r,\varphi,\psi,w)\in{\mathbf{b}}^{+}|\boldsymbol{\Phi}_{[0,\infty)}(r,\varphi,\psi,w)\subseteq\mathbf{B}\}$. On the other hand, the $\boldsymbol{\omega}$-limit set of a point $(\delta,\varphi_{0},\psi_{0},w_{\delta})\in\{(r,\varphi,\psi,w)\in{\mathbf{b}}^{+}|\boldsymbol{\Phi}_{[0,\infty)}(r,\varphi,\psi,w)\subseteq\mathbf{B}\}$ is in the collision manifold $\mathbf{N}$. This means that $\sin\varphi_{0}=0$ and $\boldsymbol{\omega}(\delta,\varphi_{0},\psi_{0},w_{\delta})=\mathbf{S}^{+}$ which yields $\varphi_{0}=0$. Thus, ${\bf a}^{+}=\{(r,\varphi,\psi,w)\in{\mathbf{b}}^{+}\,|\,\boldsymbol{\Phi}_{[0,\infty)}(r,\varphi,\psi,w)\subseteq\mathbf{B}\}$ and similarly, ${\bf a}^{-}=\{(r,\varphi,\psi,w)\in{\mathbf{b}}^{-}\,|\,\boldsymbol{\Phi}_{[-\infty,0)}(r,\varphi,\psi,w)\subseteq\mathbf{B}\}.$

Now we define the map across the block

[TABLE]

where $t_{exit}=$ the time of exit from $\mathbf{B}$. With this definition, Theorem 2.4 holds.

Consider the flow starting from a point $\mathbf{P}_{0}=(\delta,0,\psi_{0},w_{\delta})\in\mathbf{a}^{+}$. As $r\rightarrow 0$, by Lemma 5.2, $\boldsymbol{\omega(P_{0})}$ is the point $\mathbf{s}^{+}=(0,0,\psi_{0},(1/2)\ln 2)\in\mathbf{S}^{+}$. The flow emerging from $\mathbf{s}^{+}$ then follows the unstable manifold of $\mathbf{s}^{+}$. Since the unstable manifold on $\mathbf{s}^{+}$ lies in the collision manifold $\mathbf{N}$, we proceed to study the flow on $\mathbf{N}$.

As remarked before, on $\mathbf{N}$ the trajectories are given by $\psi(\tau)=\text{constant}$. Thus, the flow follows the unstable manifold of $\mathbf{s}^{+}$ to a point $\mathbf{s}^{-}=(0,\pi,\psi_{0},(1/2)\ln 2)\in\mathbf{S}^{-}$. But from Lemma 5.2, the flow follows the unstable manifold of $\mathbf{s}^{-}$ till it reaches the point $(\delta,\pi,\psi_{0},w_{\delta})\in\mathbf{a}^{-}$.

Proposition 5.5

In the logarithm problem, the map across the block extends to a homeomorphism from ${\mathbf{b}}^{+}$ to ${\mathbf{b}}^{-}$.

Proof: At $r=\delta,$ the angular momentum integral reads

[TABLE]

For $c>0$ we have $\varphi(0)\in(0,\pi)$ and at the time of exit, since $r(t_{exit})=\delta$, we must have $\varphi(t_{exit})=\varphi(0)$ or $\varphi(t_{exit})=\pi-\varphi(0).$ But from (24), $\varphi$ is increasing and so we must have $\varphi(t_{exit})=\pi-\varphi(0).$ Further, from (25), with $\varphi(0)=\varphi_{0}$, we can write,

[TABLE]

Since $r(t)$, $\varphi(t)$ and $u(t)$ so not depend on $\psi$, and their evolution is determined by their initial conditions $(\delta,\varphi_{0},w_{\delta})$, and since $\delta$ is fixed, we have $\int\limits_{0}^{t_{exit}}r^{2}(t;\delta)\sin\varphi(t;\varphi_{0})dt=G(\varphi_{0})$, where $G$ is some function. Consequently, for $c>0$ (and thus $\varphi_{0}\in(0,\pi)$) the map $\Psi$ reads

[TABLE]

Similarly, for $c<0$ (and thus $\varphi_{0}\in(\pi,2\pi)$) the map $\Psi$ reads

[TABLE]

Define $\Psi(\delta,0,\psi_{0},w_{\delta})=(\delta,\pi,\psi_{0},w_{\delta})$. Since the system is time-reversible, it suffices to show that there is continuous extension of $\Psi$ from ${\mathbf{b}}^{+}$ to ${\mathbf{b}}^{-}$. In order to prove that $\Psi$ extends to a continuous map ${\mathbf{b}}^{+}\rightarrow{\mathbf{b}}^{-}$, we need to show that $\lim_{\varphi_{0}\rightarrow 0}G(\varphi_{0})=0$. First note that, $\lim_{\varphi_{0}\rightarrow 0}r^{2}(t;\delta)\sin\varphi(t;\varphi_{0})=r^{2}(t;\delta)\sin\varphi(t;0)$, since the vector field in (23) - (26) is smooth. This means that $(r(0),\varphi(0),\psi(0),u(0))=(\delta,0,\psi_{0},w_{\delta})\in{\bf a}^{+}$. But then, $r\rightarrow 0$ as $\tau\rightarrow\infty$. This means, by Lemma 5.3, that $\sin\varphi\equiv 0$. Hence, $\lim_{\varphi_{0}\rightarrow 0}r^{2}(t;\delta)\sin\varphi(t;\varphi_{0})=0$. Note that since $r\leq\delta$, it follows that $|G(\varphi_{0})|\leq 2\delta^{2}$. Thus, by using the Lebesgue Dominated Convergence Theorem and noting that since the flow starting from a point in ${\bf a}^{+}$ remains in the block, the time of exit for such a point is infinity, we obtain,

[TABLE]

which completes the proof.

$\square$

Remark 5.6

In the case of homogeneous potentials of the form

[TABLE]

*the map across the block is extended to a diffeomorphism by removing the sets ${\bf a}^{+}$ and ${\bf a}^{-}$ through the use of a generalisation of the Levi-Civita transformation and rescaling the energy [McGe81]. Specifically, identifying the plane of the motion with the complex plane, this is achieved with the help of the conformal transformation $z\mapsto z^{n}$. (For more on conformal transformations applied to planar central force problems, see [GBM08].) Our attempts in finding a regularising conformal transformation in the case of the logarithm potential were not successful; this problem remains open. *

6 Acknowledgements

This work was partially supported by a NSERC Discovery Grant.