Yet Another Class of New Solvable N-Body Problems of Goldfish Type

Abstract

A new class of solvable N-body problems of goldfish type is exhibited and the properties of their solutions are tersely discussed. It is moreover shown that these are integrable Hamiltonian systems.

Appendix: the two-body case

In this Appendix we display the equations of motion of our model, and we outline phenomenological aspects of their solutions, in the simplest case, that with $N=2$. The quantities used below are always defined as in Proposition 2.1 (but for the convenience of the reader we occasionally recall below their definitions).

The equations of motion (3a) with $N=2$ read as follows:

$$\begin{aligned} \ddot{z}_{n}= & {} \left[ z_{n}-z_{\tilde{n}\left( n\right) }\right] ^{-1}~\left\{ 2~\dot{z}_{n}~\dot{z}_{\tilde{n}\left( n\right) }\right. \nonumber \\&\left. -\left( \omega _{1}\right) ^{2}~\left[ z_{1}+z_{2}-\left( \lambda _{1}\right) ^{2}~\left( z_{1}+z_{2}\right) ^{-3}\right] ~z_{n}\right. \nonumber \\&\left. +\left( \omega _{2}\right) ^{2}~\left[ z_{1}~z_{2}-\left( \lambda _{2}\right) ^{2}~\left( z_{1}~z_{2}\right) ^{-3}\right] \right\} , \nonumber \\ {\tilde{n}}(1)= & {} 2~,~{\tilde{n}}(2)=1,\quad n=1,2. \end{aligned}$$

(13)

The explicit solution of these equations of motions reads as follows:

$$\begin{aligned} z_{n}\left( t\right) =-\frac{1}{2}~\left[ c_{1}\left( t\right) +\left( -1\right) ^{n}~\left\{ \left[ c_{1}\left( t\right) \right] ^{2}-4~c_{2}\left( t\right) \right\} ^{1/2}\right] ,\quad n=1,2, \end{aligned}$$

(14)

with $c_{1}(t)$ and $c_{2}(t)$ given, in terms of the initial data $z_{1}(0)$, $z_{2}(0)$, by the formulas (4) with $n=1,2$ where

$$\begin{aligned} c_{1}\left( 0\right) =-\left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] ,\quad c_{2}\left( 0\right) =z_{1}\left( 0\right) ~z_{2}\left( 0\right) . \end{aligned}$$

(15)

Of course the necessary and sufficient condition for this solution to be nonsingular is validity for all time of the inequality

$$\begin{aligned} \left[ c_{1}\left( t\right) \right] ^{2}>4~c_{2}\left( t\right) . \end{aligned}$$

(16)

Let us consider firstly the more generic case with $\omega _{m}>0$ and $\omega _{m}\lambda _{m}=\mu _{m}=\mu _{m}>0$ for $m=1,2$. Then (see Remark 2.1) the initial data must of course satisfy the requirements $c_{1}(0) \ne 0$ and $c_{2}(0) \ne 0$ implying $z_{1}(0) \ne 0$, $z_{2}(0) \ne 0$ , $z_{1}(0) \ne -z_{2}(0)$ (see (15)). It is then easily seen that $c_{1}(t) \ne 0$ and $c_{2}(t) \ne 0$ for all time, implying that $c_{2}(t) =z_{1}(t) ~z_{2}(t)$—hence also $z_{1}(t)$ and $z_{2}(t)$—do not change sign over time. Hence if $z_{1}(0) <0$ and $z_{2}( 0) >0$ or viceversa $z_{1}(0) >0$ and $z_{2}(0) <0$—implying $c_{2}(t) <0$—the solution (14) is nonsingular; note the open character of this set of initial data, yielding a solution in which the two particles are for all time on opposite sides of the origin of the real line and oscillate multiperiodically—their motions being a nonlinear superpositions of two periodic evolutions with different periods $T_{1}=\pi /\omega _{1}$ and $T_{2}=\pi /\omega _{2}$—which however, if the ratio $\omega _{1}/\omega _{2}$ is a rational number, is overall periodic, implying then that the model is isochronous, all its nonsingular solutions being completely periodic with a fixed period independent of the initial data. Note that in this case there are 4 different equilibrium configurations,

$$\begin{aligned} z_{n}={\bar{z}}_{n}=\frac{1}{2}~\left\{ s_{1}\left| \left( \lambda _{1}\right) ^{1/2}\right| +s_{2}~\left( -1\right) ^{n}~\left[ \lambda _{1}+4\left| \left( \lambda _{2}\right) ^{1/2}\right| \right] ^{1/2}\right\} ,\quad n=1,2, \nonumber \\ \end{aligned}$$

(17a)

with $s_{1}=\pm$ and $s_{2}=\pm$ (but of course two of these configurations are trivially related to each other by an exchange of the labels of the two particles).

If instead initially the two particles are on the real line on the same side of the origin—i. e., $z_{1}(0)$ and $z_{2}(0)$ are both positive or both negative—then provided $\lambda _{1}^{2}>16\lambda _{2}$ there still is an open set of initial data yielding solutions which are nonsingular, those in which the two particles have initially sufficiently small velocities and are sufficiently close to the equilibrium configurations

$$\begin{aligned} z_{n}={\bar{z}}_{n}=\frac{1}{2}~\left\{ s_{1}\left| \left( \lambda _{1}\right) ^{1/2}\right| \pm \left( -1\right) ^{n}~\left[ \lambda _{1}-4\left| \left( \lambda _{2}\right) ^{1/2}\right| \right] ^{1/2}\right\} ,\quad n=1,2, \quad \quad \end{aligned}$$

(17b)

with $s_{1}=1$ if the two particles are (initially hence always) to the right of the origin (i. e., $z_{1}>0$, $z_{2}>0$) and $s_{1}=-1$ in the opposite case ($z_{1}<0$, $z_{2}<0$). Indeed, it is not difficult to verify that a condition on the initial data sufficient to guarantee that the corresponding motion be nonsingular is

$$\begin{aligned} \left( \lambda _{1}\right) ^{2}+16~h_{2}>8~h_{1}~\left| \left( h_{2}\right) ^{1/2}\right| , \end{aligned}$$

(17c)

with $h_{1}$ and $h_{2}$ defined in terms of the initial data by (4d).

Next let us consider the case with both parameters $\omega _{m}$ again positive, $\omega _{1}>0$ and $\omega _{2}>0$, but instead both parameters $\omega _{m}\lambda _{m}=\mu _{m}=\mu _{m}$ vanishing, $\omega _{1}\lambda _{1}=\omega _{2}\lambda _{2}=\mu _{m}=0$. In this case the only equilibrium position is characterized by $c_{m}=c_{m}=0$ for $m=1,2$ , see (4j), and the nonsingular solutions are quite nongeneric; they clearly may only exist if $\omega _{1}$ and $\omega _{2}$ are congruent (i. e., integer multiples of each other), and even in such cases they require quite special sets of initial data. For instance, a simple such case would be that with $\omega _{2}=2\omega _{1}$; then the model would of course be isochronous, with all solutions completely periodic with period $2\pi /\omega _{2}$; but conditions on the initial data sufficient to guarantee that the solution be nonsingular would require the initial data to imply validity of the equality $\theta _{2}=2\theta _{1}$ $\mathrm{mod} ( 2\pi )$ and of the inequality $( A_{1})^{2}>4A_{2}$, see (4j).

Next, let us consider the two cases with both parameters $\omega _{m}$ again positive, $\omega _{1}>0$ and $\omega _{2}>0$, but only one of the two parameters $\omega _{m}\lambda _{m}=\mu _{m}=\mu _{m}$ vanishing. If $\omega _{1}\lambda _{1}=\mu _{1}>0$, $\omega _{2}\lambda _{2}=\mu _{2}=0$, the 4 equilibrium configurations of the system are $z_{1}=0$, $z_{2}=\pm \vert ( \lambda _{1})^{1/2}\vert$ and (of course) $z_{1}=\pm \vert ( \lambda _{1})^{1/2}\vert$, $z_{2}=0$; and the nonsingular solutions would be characterized by initial data $z_{1}(0)$, $z_{2}( 0)$ sufficiently close to these equilibria and with sufficiently small (in modulus) initial velocities $\dot{z}_{1}(0)$, $\dot{z}_{2}(0)$, featuring coordinates $z_{n}(t)$ which oscillate around their equilibrium values, multiperiodically if $\omega _{1}/\omega _{2}$ is not a rational number, periodically otherwise (isochronous case). If instead $\omega _{1}\lambda _{1}=\mu _{1}=0$, $\omega _{2}\lambda _{2}=\mu _{2}>0$, all solutions characterized by initial data featuring the two particles initially on different sides of the origin, so that $c_{2}(0) =z_{1}(0)$ $z_{2}(0) <0,$ would be nonsingular, maintaining this feature throughout their time evolution—which would again be multiperiodic if $\omega _{1}/\omega _{2}$ is not a rational number, periodic otherwise (isochronous case). In this case the two equilibria around which the two particles oscillate are $z_{1}=s\vert ( \lambda _{2})^{1/4}\vert$, $z_{2}=-s\vert ( \lambda _{2})^{1/4}\vert$ with $s=\pm$.

Next, let us consider the cases when both parameters $\omega _{m}$ vanish, $\omega _{1}=\omega _{2}=0$ (excluding however the well-known case with $\omega _{1}\lambda _{1}=\omega _{2}\lambda _{2}=0$, corresponding to the original goldfish model). In these cases no periodic solutions exist: at least one particle comes in from infinitely far away in the remote past and one escapes infinitely far away in the remote future. In the case $\omega _{2}\lambda _{2}=\mu _{2}>0$ (of course with $\omega _{1}\lambda _{1}=0$) if the two particles are initially on opposite sides of the origin, so that $c_{2}(0) =z_{1}(0) z_{2}(0) <0,$ then the motions are nonsingular for all initial data; each particle remains on the same side of the origin where it was initially, one having come from infinitely far away in the remote past and approaching asymptotically the origin in the remote future, the other having been asymptotically close to the origin in the remote past and escaping to infinity in the remote future; in formulas, if $c_{1}(0) \ne 0$ ,

$$\begin{aligned} z_{s}\left( t\right) =-\frac{1}{2}\left\{ s~\left| c_{1}\left( 0\right) ~t+c_{1}\left( 0\right) \right| +c_{1}\left( 0\right) ~t+c_{1}\left( 0\right) \right\} +O\left( \left| \frac{1}{t}\right| \right) ,\quad s=\pm ,\nonumber \\ \end{aligned}$$

(18a)

where we denote as $z_{+}(t)$ respectively as $z_{-}(t)$ the coordinate of the particle that goes towards infinity respectively towards the origin as $c_{1}( 0) ~t\rightarrow +\infty$; here of course $c_{1}( 0) =-[ z_{+}(0) +z_{-}(0] ,$ $c_{1}(0) =-[ \dot{z}_{+}(0) +\dot{z}_{-}(0] ,$ $c_{2}(0) =z_{+}(0) z_{-}(0) <0$. While if $c_{1}(0) =0,$ then the behavior is different, with both particles coming in from far away in the remote past and returning far away in the remote future, of course each remaining for all time on the same side of the origin. In formulas the asymptotic behavior in this case reads as follows (as $t\rightarrow \pm \infty )$

$$\begin{aligned} z_{s}\left( t\right)= & {} 2~s\left| \left\{ \left( \mu _{2}\right) ^{2}~ \left[ z_{1}\left( 0\right) ~z_{2}\left( 0\right) \right] ^{-2}+\left[ \dot{z }_{1}\left( 0\right) ~z_{2}\left( 0\right) +z_{1}\left( 0\right) ~\dot{z}_{2}\left( 0\right) \right] ^{2}\right\} ^{1/2}~t\right| ^{1/2} \nonumber \\&+\frac{1}{2}~\left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] +O\left( \left| t\right| ^{-1/2}\right) ,\quad s=\pm , \end{aligned}$$

(18b)

where now $z_{+}(t)$ respectively $z_{-}(t)$ is the coordinate of the particle on the right respectively on the left.

On the other hand if $\omega _{1}\lambda _{1}=\mu _{1}>0$ (while of course $\omega _{2}\lambda _{2}=\mu _{2}=0$) then as $t\rightarrow \pm \infty$

$$\begin{aligned} z_{+}\left( t\right)= & {} \left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] \left| ~\left| \left( 1-\rho _{1}^{2}\right) ^{1/2}\right| +\frac{\mu _{1}~t}{\rho _{1}~\left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] ^{2}}\right| \nonumber \\&+\,J+O\left( \left| t\right| ^{-1}\right) ,\quad z_{-}\left( t\right) =-J+O\left( \left| t\right| ^{-1}\right) , \nonumber \\ J= & {} \frac{\left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] ~\left[ \dot{z}_{1}\left( 0\right) ~z_{2}\left( 0\right) +z_{1}\left( 0\right) ~\dot{ z}_{2}\left( 0\right) \right] }{\left| \left( \mu _{1}\right) ^{2}+\left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] ^{2}~\left[ \dot{z}_{1}\left( 0\right) +\dot{z}_{2}\left( 0\right) \right] ^{2}~\right| ^{1/2}}, \end{aligned}$$

(19)

where the label “$+$” denotes the particle that comes in from far away in the remote past and returns back far away in the remote future, while the other particle (identified with the label “−”) always remains in a finite part of the real line. Note that in this case the possibility of a collision cannot be a priori excluded; precise conditions stating whether, for given parameters and initial data, this does or does not happen can be obtained, but they are not sufficiently enlightening to be worth reporting; it is however plain that the nonsingular solutions (featuring no collision) emerge also in this case from an open set of initial data.

Finally, let us consider the cases in which one of the two parameters $\omega _{m}$ is positive and the other one vanishes.

Firstly let us consider the cases with $\omega _{1}>0,$ $\omega _{2}=0$. Let us then assume, firstly, that $\omega _{2}\lambda _{2}=\mu _{2}>0$ (while $\omega _{1}\lambda _{1}=\mu _{1}\ge 0$). It is then plain that the initial data must satisfy the restriction $z_{1}(0) \ne 0,$ $z_{2}(0) \ne 0$ (and, if $\omega _{1}\lambda _{1}>0,$ also $z_{1}(0) \ne -z_{2}(0)$), and that the coordinates $z_{n}( t)$ cannot change sign throughout the time evolution. Then the necessary and sufficient condition for all solutions of the model to be nonsingular is that the two particles be on different sides of the origin. And these nonsingular solutions are characterized by each particle coming in from far away in the remote past and returning far away in the remote future. If instead $\omega _{2}\lambda _{2}=\mu _{2}=0$, then conditions on the initial data sufficient to insure that the solutions be nonsingular for $t\ge 0$ are the two inequalities

$$\begin{aligned} z_{1}\left( 0\right) ~z_{2}\left( 0\right)<0,\quad \dot{z}_{1}\left( 0\right) ~z_{2}\left( 0\right) +\dot{z}_{2}\left( 0\right) ~z_{1}\left( 0\right) <0, \end{aligned}$$

(20)

with the corresponding solutions again featuring two particles flying away, one to $+\infty$ and the other to $-\infty$ as $t\rightarrow +\infty$; but these solutions would be singular at some negative time (unless the second inequality (20) became an equality, making the initial data nongeneric; the corresponding solution would then oscillate periodically in the finite part of the real line, with period $T_{1}=\pi /\omega _{1}$ if $\lambda _{1}>0,$ $2T_{1}$ if $\lambda _{1}=0$).

Next let us consider the cases with $\omega _{1}=0,$ $\omega _{2}>0$. Then if $\omega _{2}\lambda _{2}=\mu _{2}>0$ the origin is a forbidden point for the particles, initially and throughout the motion, hence no particle can cross the origin; therefore all solutions featuring the two particles on opposite sides of the origin are nonsingular, with one particle coming in from infinitely far away in the remote past and returning there in the remote future (with asymptotically constant velocities), and the other one approaching asymptotically the origin with decreasing velocity (but of course never reaching it); unless $\lambda _{1}$ is finite (so that $\omega _{1}\lambda _{1}=0$) and moreover the (nongeneric) initial data imply $c_{1}(0) =0$—hence $\dot{z}_{1}(0) =-\dot{z}_{2}(0)$ —when the solution is periodic with period $T_{2}=\pi /\omega _{2}$. If the particles are instead on the same side of the origin, then all generic solutions are singular at some time, but there is an open set of initial data—characterized by one inequality (which is not sufficiently enlightening to justify explicit display)—which is sufficient to guarantee that the solutions are nonsingular for $t\ge 0$. The asymptotic behavior as $t\rightarrow +\infty$ of these solutions sees one coordinate (denoted below as $z_{+}( t)$) escaping far away in the remote future and the other one (denoted below as $z_{-}( t)$) tending instead to a finite value. In formulas, if $\omega _{1}\lambda _{1}=\mu _{1}>0$ then

$$\begin{aligned} z_{+}\left( t\right)= & {} -\frac{\mu _{1}~t}{\rho _{1}~c_{1}\left( 0\right) } +E+O\left( \frac{1}{t}\right) ,\quad z_{-}\left( t\right) =-E+O\left( \frac{1}{ t}\right) , \nonumber \\ E= & {} \frac{\rho _{1}~\mu _{2}~c_{1}\left( 0\right) }{\rho _{2}~\mu _{1}~c_{2}\left( 0\right) },\quad c_{1}\left( 0\right) \ne 0,\quad c_{2}\left( 0\right) \ne 0; \end{aligned}$$

(21a)

if instead $\omega _{1}\lambda _{1}=\mu _{1}=0,$ provided $c_{1}(0) c_{1}(0) >0$, then

$$\begin{aligned} z_{+}\left( t\right)= & {} -\left[ c_{1}\left( 0\right) ~t+c_{1}\left( 0\right) \right] +F+O\left( \frac{1}{t}\right) ,\quad z_{-}\left( t\right) =-F+O\left( \frac{1}{t}\right) , \nonumber \\ F= & {} \frac{\mu _{2}~}{\rho _{2}~\left| c_{1}\left( 0\right) \right| ~c_{2}\left( 0\right) },\quad c_{1}\left( 0\right) \ne 0,\quad c_{2}\left( 0\right) \ne 0. \end{aligned}$$

(21b)

Here of course we have used the notation of Proposition 2.1, see (4).

The last cases to be considered are those with $\omega _{1}=0,$ $\omega _{2}>0$ and $\omega _{2}\lambda _{2}=\mu _{2}=0$. Then—if $\omega _{1}\lambda _{1}=\mu _{1}=0$—the solution is certainly nonsingular for $t\ge 0$—while it is likely to feature a singularity at some negative value of t—provided

$$\begin{aligned}&\left[ \dot{z}_{1}\left( 0\right) +\dot{z}_{2}\left( 0\right) \right] ~ \left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right]>0,\quad \left[ z_{1}\left( 0\right) +z_{2}\left( 0\right) \right] ^{2} \nonumber \\&\quad >4~\left| \left\{ \left[ z_{1}\left( 0\right) ~z_{2}\left( 0\right) \right] ^{2}+\left[ \frac{\dot{z}_{1}\left( 0\right) ~z_{2}\left( 0\right) +z_{1}\left( 0\right) ~\dot{z}_{2}\left( 0\right) }{\omega _{2}}\right] ^{2}\right\} ^{1/2}\right| , \end{aligned}$$

(22a)

featuring then, as $t\rightarrow +\infty$, one particle traveling infinitely far away (to the left if $z_{1}(0) +z_{2}(0) <0,$ to the right if $z_{1}(0) +z_{2}(0) >0$ ) and the other one approaching asymptotically the origin. While if $\omega _{1}\lambda _{1}=\mu _{1}>0$ a condition on the initial data sufficient to guarantee that the solutions be nonsingular for all time is (in the notation of Proposition 2.1, see (4))

$$\begin{aligned} \left[ c_{1}\left( 0\right) \right] ^{2}~\left( \rho _{1}\right) ^{2}>4~A_{2}; \end{aligned}$$

(22b)

and in this case one coordinate comes in from far away in the remote past and approaches the origin in the remote future, while the other goes far away (on the opposite side) in the remote future and was instead near the origin in the remote past.

Let us end this Appendix by displaying the Hamiltonian that yields the equations of motion (13):

$$\begin{aligned} H\left( x_{1},~x_{2};~z_{1},~z_{2}\right)= & {} \frac{1}{2}~\left\{ \left( z_{1}-z_{2}\right) ^{-2}~\left[ \left( \frac{x_{1}-x_{2}}{a}\right) ^{2}+\left( \frac{x_{1}~z_{1}-x_{2}~z_{2}}{a}\right) ^{2}\right] \right. \nonumber \\&\left. +\left( \omega _{1}\right) ^{2}~\left[ \left( z_{1}+z_{2}\right) ^{2}+\left( \lambda _{1}\right) ^{2}~\left( z_{1}+z_{2}\right) ^{-2}\right] \right. \nonumber \\&\left. +\left( \omega _{2}\right) ^{2}~\left[ \left( z_{1}~z_{2}\right) ^{2}+\left( \lambda _{2}\right) ^{2}~\left( z_{1}~z_{2}\right) ^{-2}\right] ~\right\} , \end{aligned}$$

(23)

where of course $x_{1}\equiv x_{1}(t)$ respectively $x_{2}\equiv x_{2}(t)$ are the canonical momenta conjugated to the canonical coordinates $z_{1}\equiv z_{1}( t)$ respectively $z_{2}\equiv z_{2}(t)$ (while a is an arbitrary constant the role of which is merely to adjust the dimensions: it could be absorbed in the definition of the canonical momenta or equivalently rescaled away by renormalizing appropriately the time variable).