A dissipative Kepler problem with a family of singular drags

Abstract

In this work, we consider the Kepler problem with a family of singular dissipations of the form $-\frac{k}{|x|^\beta }\dot{x},\quad k>0, \beta >0.$ We present some results about the qualitative dynamics as $\beta$ increases from zero (linear drag) to infinity. In particular, we detect some threshold values of $\beta$, for which qualitative changes in the global dynamics occur. In the case $\beta =2$, we refine some results obtained by Diacu and prove that, unlike for the case of the linear drag, the asymptotic Runge–Lenz vector is discontinuous.

Appendix

Proof

(of Theorem7) This proof borrows some ideas from the ones of Proposition 2.4 in Margheri et al. (2017b) and of Proposition 3.1 in Margheri et al. (2014), mainly in Case I. The corresponding steps are presented below with less detail.

We recall that our phase space is the half-plane $\Omega ^0=\{(r,u): \ r>0, u\in \mathbb R \}.$ The isocline of system (34) associated with $\dot{r}=0$ is the half line defined by $u=0,$ whereas for $\dot{u}=0$ the isocline is defined by

$$\begin{aligned} u=-\frac{r^{\beta -2}}{k}. \end{aligned}$$

These curves determine the following disjoint open regions in the phase space:

$$\begin{aligned} \begin{aligned} A_0&=\{ (r,u)\in \Omega ^0: \ u> 0\},\quad A_1=\left\{ (r,u)\in \Omega ^0: \ 0> k\,u>-r^{\beta -2} \right\} , \\ A_2&=\left\{ (r,u)\in \Omega ^0: \ k\,u<-r^{\beta -2} \right\} . \end{aligned} \end{aligned}$$

(44)

The set $A_0$ is negatively invariant with respect to the flow of (34) for all $\beta >0$, whereas for $\beta \in ]0,2[$, $A_1$ is positively invariant and $A_2$ is negatively invariant. For $\beta >2,$ the set $A_2$ is positively invariant. We distinguish three cases.

Case 1.$\beta \in ]0,2[$.

We prove first that $\omega$ is finite. If it were infinite, there should exist a sequence $t_n\rightarrow +\infty$ such that $u(t_n)\rightarrow 0$. However, this is not possible, because one can easily check that all collision solutions enter eventually into the positively invariant set $A_1,$ where $u=\dot{r}$ is negative and decreasing.

In what follows, we consider the regularization of system (34) given by the time rescaling $d\mu =\frac{\mathrm{d}t}{r^2},$ already considered in the proof of Theorem 5. Of course, we obtain a system made by the first two equations of (24) with $\varphi \equiv 0$, namely,

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\mathrm{d}r}{\mathrm{d}\mu }=r^2 u, \\ \frac{\mathrm{d}u}{\mathrm{d}\mu }=-k\, r^{2-\beta } u - 1. \end{array} \right. \end{aligned}$$

(45)

System (45) is defined in the extended phase space $\bar{\Omega }^0=\Omega ^0\cup \{(r,u): \ r=0, u\in \mathbb R \}$, and the line $r=0$ is an isocline orbit associated with $\frac{\mathrm{d}r}{\mathrm{d}\mu }=0$. (See the left panel of Fig. 2.)

Fig. 2

Dependence on $\beta$ of the regions defined by the isoclines in the regularized systems

Let us prove now that, when $\beta \in ]0,2[$, the velocity at collision satisfies $u_\omega =-\infty .$ We argue by contradiction. Assume that $\mu \mapsto (r(\mu ),u(\mu ))$ is a collision solution such that $u_\omega \in ]-\infty ,0[$. We can take initial conditions $(r(0),u(0))\in A_1$ so that $u(\mu )$ is negative for all $\mu >0$. Then, from the first equation of (45), we see that

$$\begin{aligned} \frac{1}{r(\mu )}=\frac{1}{r(0)}+\int _0^\mu |u(\sigma )| \mathrm{d} \sigma . \end{aligned}$$

(46)

Letting $\mu \rightarrow \mu _\omega$, we have that the left-hand side tends to $+\infty ,$ and since $u(\tau )$ is bounded on $[0, \mu _\omega [,$ from (46) we get $\mu _\omega =+\infty$. But then, from the second equation of (45), it follows that

$$\begin{aligned} \lim _{\mu \rightarrow +\infty } \frac{\mathrm{d}u}{\mathrm{d}\mu } = -1, \end{aligned}$$

and this would imply that $u_\omega =-\infty$, contradicting the hypothesis that $u_\omega$ is finite. We conclude that $u_\omega =-\infty .$

Now, let us see how the energy behaves when an orbit approaches the collision.

Consider first the case $\beta \in ]0,1[.$ System (45) has the first integral

$$\begin{aligned} H(r,u,\mu )= u +\frac{k}{1-\beta } r^{1-\beta } + \mu , \end{aligned}$$

(47)

obtained by setting $\varphi \equiv 0$ in (25).

Let $(r_0,u_0)$ be an initial condition in $A_1$,  and let $H_0 := H(r_0,u_0,0),$ so that by (47) we have

$$\begin{aligned} u(\mu )+\frac{k}{1-\beta }r^{1-\beta }(\mu )+\mu =H_0. \end{aligned}$$

(48)

Since $u(\mu )\rightarrow -\infty$ and $r(\mu )\rightarrow 0$ as $\mu \rightarrow \mu _\omega$, from (48) we infer that $\mu _\omega = +\infty .$ Then, from the same equality it follows that $\frac{u(\mu )}{\mu } \rightarrow -1$ as $\mu \rightarrow +\infty ,$ and from (46) we get that $\mu ^2r(\mu ) \rightarrow 2$ as $\mu \rightarrow +\infty .$

By (4), we see that $\frac{\mathrm{d}E}{\mathrm{d}\mu }=r^2\frac{\mathrm{d}E}{\mathrm{d}t}=-k\, r^{2-\beta } u^2.$ Then, using the two limits established above for u and r, we get that

$$\begin{aligned} \mu ^{2(1-\beta )}\frac{\mathrm{d}E}{\mathrm{d}\mu } (\mu ) \rightarrow -2^{2-\beta } k, \end{aligned}$$

as $\mu \rightarrow \infty$. As a consequence,

$$\begin{aligned} E_\omega = E(0)+\int _0^\infty \frac{\mathrm{d}E}{\mathrm{d}\mu } (\sigma ) \mathrm{d}\sigma \end{aligned}$$

is finite if $\beta \in ]0,1/2[,$ whereas $E_\omega =-\infty$ if $\beta =[1/2,1[$.

Let us consider now $\beta \in [1,2[.$

In this case, the approach through the first integral (given by $H=u+k\log r +\mu ,$ when $\beta =1,$ and by (47), when $\beta >1$) does not allow to find out the asymptotic expansion of the solutions as they approach collision. However, we argue as follows. It is easy to see that in $A_1$ the trajectories of system (45) may be written in the form $u=\chi (r), \,\,r\in ]0,r_0].$ If we evaluate the slope of such orbits at the points of the form $u=-r^{\beta -1},$ we get

$$\begin{aligned} \left. \frac{\mathrm{d}u}{\mathrm{d}r} \right| _{u=-r^{\beta -1}} = \frac{1}{r^\beta } \left( { \frac{1}{r} -k}\right) , \end{aligned}$$

which is positive for $0<r<1/k.$ This implies that the region

$$\begin{aligned} D:=\left\{ (r,u): 0<r<\frac{1}{k},\,\, -\frac{1}{kr^{2-\beta }}<u< -r^{\beta -1} \right\} \subset A_1 \end{aligned}$$

is positively invariant. On the other hand, note that

$$\begin{aligned} \frac{\mathrm{d}E}{\mathrm{d}r} = \frac{1}{u} \frac{\mathrm{d}E}{\mathrm{d}t} = -k\frac{u}{r^\beta }>0. \end{aligned}$$

Then, for every orbit $u=\chi (r)$ such that $(r_0,\chi (r_0))\in D$, as a consequence of the invariance of D,  we have

$$\begin{aligned} \frac{k}{r}< \left. \frac{\mathrm{d}E}{\mathrm{d}r}\right| _{u=\chi (r)} < \frac{1}{r^2}, \end{aligned}$$

and we conclude that

$$\begin{aligned} E_\omega = E(0)+\int _{r_0}^0 \left. \frac{\mathrm{d}E}{\mathrm{d}r}\right| _{u=\chi (r)} \mathrm{d}r=-\infty . \end{aligned}$$

Case 2.$\beta =2$.

This case is solved in Diacu (1999). The explicit solution is given by

$$\begin{aligned} u(\mu ) = ({u_0+\frac{1}{k}}) e^{-k\mu } -\frac{1}{k},\qquad \frac{1}{r(\mu )}= \frac{1}{r(0)}-\int _0^\mu u(\sigma ) \mathrm{d} \sigma ,\qquad \mu \in [0,\infty [. \end{aligned}$$

From these expressions, it is easy to see that $\omega <+\infty ,$$u_\omega =-1/k$, and $E_\omega =-\infty$.

Case 3.$\beta >2$.

In order to study the collisions for $\beta > 2$, it is convenient to consider the time rescaling $d\tau =\frac{\mathrm{d}t}{r^\beta }$, introduced previously to get system (26). We obtain the following regular system, which corresponds to the first two equations of (26) with $\varphi \equiv 0,$

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\mathrm{d}r}{\mathrm{d}\tau }=r^\beta u \\ \frac{\mathrm{d}u}{\mathrm{d}\tau }=-k\, u - r^{\beta -2}, \end{array} \right. \end{aligned}$$

(49)

on the extended phase space $\bar{\Omega }^0$.

We note that the origin is an equilibrium of system (49) which attracts the points of the invariant line $r=0$. (See the right panel of Fig. 2.)

One can see easily that all collision orbits will enter eventually in the positively invariant region $A_2.$ Then, without loss of generality, we will consider only initial conditions in $A_2.$ Actually, since in this region we have $\frac{\mathrm{d}u}{\mathrm{d}\tau }>0$ and $\frac{\mathrm{d}r}{\mathrm{d}\tau }<0$, all solutions are bounded on $[0, \tau _\omega [.$ This fact implies that all solutions starting in $A_2$ are collision ones, since otherwise an equilibrium of the system should exist in $A_2.$ Moreover, any segment of orbit contained in $A_2$ may be expressed in the form $u=\chi (r),$ with r in a suitable interval of the form $]0, \tilde{r}[,$ with $0<r_0<\tilde{r}.$ Notice that, on this interval, $\chi (r)$ satisfies the scalar differential equation

$$\begin{aligned} \frac{\mathrm{d}u}{\mathrm{d}r}:=f(r,u)=-\frac{k}{r^{\beta }}-\frac{1}{r^2u}. \end{aligned}$$

(50)

When $\beta \ge 3$ the vector field associated with (49) is continuously differentiable on $\bar{\Omega }^0$ and, by the general theory of ODEs, we conclude that all solutions starting in $A_2$ tend to the equilibrium (0, 0) in infinite $\tau$ time. We conclude that $u_\omega =0.$

This cannot be guaranteed without further considerations for $\beta \in ]2,3[.$ In fact, in this range of values the regularized vector field is not Lipschitz continuous in the points of the form $(0,u)\in \bar{\Omega }^0.$ As a consequence, in such points uniqueness of solutions may fail, and all we can conclude by the general theory is that, for collision solutions, we have $-\infty < u_\omega \le 0.$ Let us prove that, actually, $u_\omega =0$. The proof can be followed with Fig. 3. Define the function $h_\lambda (r):=-\lambda r^{\beta -2}$, where $\lambda >1/k.$ Note that $A_2=\cup _{\{\lambda>1/k\}}\{(r,h_\lambda (r)): r>0\}.\,$ Evaluating the slope field du/dr at $u=h_\lambda (r),$ we get

$$\begin{aligned} \left( {\left. \frac{\mathrm{d}u}{\mathrm{d}r} \right| _{h_\lambda } }\right) \bigg / \frac{\mathrm{d}h_\lambda }{\mathrm{d}r} = \frac{k\lambda - 1 }{\lambda ^2 (\beta -2) r^{2\beta -3}}. \end{aligned}$$

(51)

Fig. 3

Illustration of the proof for $\beta \in ]2,3[.$

Hence, for any fixed $\lambda >1/k\,$ there exists only one point $r=r_\lambda$ such that the graph of the function $h_\lambda (r)$ is tangent to an orbit and is given by

$$\begin{aligned} r_\lambda := \left( { \frac{k\lambda - 1 }{\lambda ^2 (\beta -2) } }\right) ^{\frac{1}{2\beta -3}}. \end{aligned}$$

(52)

Moreover, by (51) it follows that

$$\begin{aligned} \frac{\mathrm{d}h_\lambda }{\mathrm{d}r}\lessgtr f(r,h_\lambda (r)),\,\,\,\,\,\hbox {if}\,\, \,\,\, r\gtrless r_\lambda . \end{aligned}$$

(53)

As a consequence, by the comparison theorem for ODEs, the orbit $u=\chi (r)$ of (50) that is tangent to the curve $u=h_\lambda (r)$ in $r=r_\lambda$ satisfies

$$\begin{aligned} \chi (r)\ge h_\lambda (r), \end{aligned}$$

for any $r\in ]0, r_\lambda [.$

Note that (52) is a second-degree equation in $\lambda .$ Solving it, we obtain two local inverses of the function $\lambda \mapsto r_\lambda ,$ namely the functions

$$\begin{aligned} \lambda _\pm (r):=\frac{1}{r^{2\beta -3}} \frac{k\pm \sqrt{ k^2 - 4(\beta -2)r^{2\beta -3} }}{2(\beta -2)}, \end{aligned}$$

defined for $0<r\le R$, where

$$\begin{aligned} R:=\left( {\frac{k^2}{4(\beta -2)}}\right) ^{\frac{1}{2\beta -3}}. \end{aligned}$$

Now, we use $\lambda _\pm (r)$ to construct the two following auxiliary functions:

$$\begin{aligned} h_\pm (r)=h_{\lambda _\pm (r)}(r):=-\frac{1}{r^{\beta -1}} \frac{k\pm \sqrt{ k^2 - 4(\beta -2)r^{2\beta -3} }}{2(\beta -2)}, \end{aligned}$$

also defined for $0<r\le R.$

The functions $h_+$ and $h_-$ satisfy the following properties. They are, respectively, strictly decreasing and strictly increasing on ]0, R],  and such that $h_-(r)>h_+(r)$ on ]0, R[,  with $h_-(R)=h_+(R):=u_R.$ Moreover, they have the following behavior as $r\rightarrow 0^+$:

$$\begin{aligned} h_-(r)=-\frac{r^{\beta -2}}{k} + o(r^{\beta -2}) \ \rightarrow 0 \quad \hbox {and} \quad h_+\rightarrow -\infty . \end{aligned}$$

(54)

Finally, the range of $h_+$ is $]-\infty ,u_R],$ and the one of $h_-$ is $[u_R,0[$. We are now ready to prove that on collision orbits $u_\omega =0$.

We start by defining the positively invariant set

$$\begin{aligned} G := \{(r,u): \ 0< r \le R,\ h_+(r)\le u\le h_-(r)\}\subset A_2. \end{aligned}$$

Given an initial condition $(r_0,u_0)\in G,$ let us consider the corresponding orbit $u=\chi (r), \,r\in ]0,\tilde{r}[.$ Since there exists a value $\lambda >1/k$ such that $h_-(r)>\chi (r)>-\lambda r^{\beta -2}$ for all $0<r<R,$ we conclude that the orbit will go toward the equilibrium as $r\rightarrow 0^+$. If $(r_0,u_0)\in A_2\backslash G\,$, the corresponding orbit will eventually enter in G. In fact, if there exists an orbit $u=\bar{\chi }(r)$ for which this is not the case, we can find $\bar{\lambda }=\lambda _-(\bar{r})$ such that $u=h_{\bar{\lambda }}(r)$ intersects $u=\bar{\chi }(r)$ in a point $\hat{r}>\bar{r}$ for which it holds the inequality $\frac{\mathrm{d}h_{\bar{\lambda }}}{\mathrm{d}r}(\hat{r})>f(\hat{r}, h_{\bar{\lambda }}(\hat{r})).\,$ By (53), there exists a second point, $r_*>\hat{r}>\bar{r},$ such that the curve $u=h_{\bar{\lambda }}(r)$ is tangent to an orbit (the first being $\bar{r}$), which is absurd.

Then, $u_\omega = 0$ for all collision orbits also for $\beta \in ]2,3[.$ Taking into account what was proved previously, we conclude that $u_\omega = 0$ for any $\beta >2.$ It follows immediately that the energy $E=u^2/2-1/r$ tends to $E_\omega =-\infty$.

Also, since

$$\begin{aligned} \omega = \int _{r_0}^0 \frac{\mathrm{d}r}{\chi (r)}, \end{aligned}$$

(55)

by (54) and by the inequality $-\lambda r^{\beta -2}<\chi (r)<h_-(r)\,$ for any $r\in ]0, R[,$ we see that $1/|\chi (r)|$ is integrable on the interval $]0,r_0]$ if $\beta \in ]2,3[,$ in which case $\omega$ is finite, whereas it is not integrable if $\beta \ge 3,$ and then $\omega = +\infty$. Our proof is concluded. $\square$