Space invaders and energy conservation

The energy is conserved (until the singularity).

Note, however, that it is not easy to construct a system in which the action of the forces will create such a singularity. This was an open problem (posed by Painlevé and Poincaré) for a very long time. It was solved in this paper, which shows that such a singularity can appear in the 5-body problem in celestial mechanics.

In his example, the divergence of the kinetic energy is made possible by the fact that the gravitational potential energy is not bounded below. This is discussed explicitly in the paper.

This issue is interesting to me. Here a subtle point with the notion of determinism in classical physics shows up. There are two distinct notions.

  1. Determinism means that if we know positions and velocities of all material points of a system at time $t_0$ (with respect to some reference frame), and we know the interactions (forces) acting on the system, then we know the entiere evolution of the system for $t>t_0$ and $t<t_0$ in the interval of time where the maximal solution with the said initial conditions is defined.

  2. Determinism means that if we know all the positions and velocities of the material points of all physical systems at time $t_0$, and we also know the interactions, we know the evolution of the entire universe for $t>t_0$ and $t<t_0$.

Version 1 is true in classical physics (provided the forces are sufficiently smooth as functions of time position and velocities), whereas the proposed example proves that version 2 may be false.

It is intersting to stress that in general relativistic physics the used notion of determinism is 2. It corresponds to the notion of global hyperbolicity of a spacetime, i.e., the requirment that there exists a smooth spacelike 3-hypersurfaces that meets all inextensible worldlines of material points. I realized that classical physics does not satisfy this requirement unless constraining the admitted class of forces.

Coming to the question, it is easy to see that no problems arise with energy conservation because this apparently weird behavior (space invader solution) of maximal soutions of the problem of motion may take place for conservative forces, where conservation of mechanical energy is valid.

Let us construct an easy example in one spatial dimension. Assume to deal with a material point with mass $m=1$ in an inertial reference frame with spatial cartesian coordinate $x$. Suppose that the point is subjected to the only conservative repulsive force due to the potential energy $$U(x) = - (1+x^2)^2\:.$$ We know that the mechanical energy $$E = \frac{1}{2}\left(\frac{dx}{dt}\right)^2 + U(x) \:, \tag{0}$$ must be therefore constant along the motion $x=x(t)$ of the system, solution of $$\frac{d^2x}{dt^2} = F(x(t)) = - \frac{d}{dx}|_{x=x(t)} U(x),\tag{1}$$ with initial conditions $$x(0) =x_0\:, \quad \frac{dx}{dt}|_{t=0} = v\:.\tag{2}$$ The proof of this conservation law immediately arise from (1). On the other hand, since the force $F$ is smooth, the existence and uniqueness theorem also proves that there is a unique maximal solution of the said problem (1)+(2) defined in the largest interval $(t_a,t_b) \ni 0$.

Let us focus on this unique maximal motion $x=x(t)$.

The conservation of $E$ reads $$\frac{1}{2}\left(\frac{dx}{dt}\right)^2 - (1+x(t)^2)^2 = E_0$$ where $$E_0 = \frac{1}{2}v^2 - (1+x_0^2)^2\:.$$ Let us pick out $x_0= 0$ and $v>0$ such that $E_0=0$. With this choice, $$\frac{1}{2}\left(\frac{dx}{dt}\right)^2 = (1+x(t)^2)^2$$ and thus $$\frac{dx}{1+ x^2} = \pm \sqrt{2} dt\:.$$ Therefore $$\int_{0}^{x(u)}\frac{dx}{1+ x^2} = \pm \sqrt{2} \int_{0}^t du \:,$$ namely (also using $v>0$) $$\arctan x(t) = \sqrt{2}t\:,$$ so that $$x(t) = \tan \sqrt{2}t\:, \quad t \in (-\pi\sqrt{2}, \pi \sqrt{2})$$ We have found that the left endpoints of the domain of the solution $(t_a,t_b)$ are finite but the orbit $x(t)$ varies in the whole $(-\infty, \infty)$. That is a space invader solution with conserved energy.