What is a Newtonian system? The failure of energy conservation and determinism in supertasks
The ``supertask'' (ST) is defined by infinitely many unit point masses \(P_1,P_2,\cdots\) arrayed at rest along the \(x\)-axis of space at positions \(x=\frac{1}{2}, \frac{3}{4}, \frac{7}{8}, \cdots\). At time \(t=0\) a unit point mass \(P_o\), moving at unit velocity, passes \(x=0\) and approaches the remaining masses. Infinitely many perfectly elastic collisions ensue with the unit velocity passing in succession from \(P_o\) to \(P_1\), from \(P_1\) to \(P_2\) etc. However, at \(t=1\), all collisions are completed, and all masses are at rest. The mathematical description of the process can be found in chapter 3 (``Choosing a mathematical model for the supertask''), which reveals the results of an assumed dynamical process without explaining the process itself explicitly: In eq. (2.1) (Newton's second law), a distiction between \(F_i=F_i^i+F_i^r\) (impressed forces and reaction forces) for the beginning of a contact is missing. With an interpretation of eq. (2.2) as d'Alembert's (original) principle (i.e. the reaction forces \(F_i^r\) equilibrate), the underlying dynamical process might be looked at as implicitly defined with \(F_i^i=0\) for all \(i\) and \(\sum F_i^r = 0\). The system under consideration moves then due to an initial condition (\(P_o\)) and undergoes constraints leading to ``instantaneous velocity changes''. These would, however, already ``threaten the Newtonian credentials'' because of its discontinuity. ``The most natural solution is to assume that each collison is mediated by very strong short range forces that eliminate the discontinuities in the time dependence of velocities and provide them sufficient differentiability for the acceleration to be always defined''. In that case, eq. (2.2) may be interpreted in the authors' sense (``Newton's third law'') as a summation over the impressed forces \(\sum F_i^i = 0\) which indicates that the impressed forces are inner forces and no additional constraints occur. D'Alembert's principle then turns to triviality and eq. (2.1) holds for \(F_i=F_i^i\). The dynamical process remains still undefined due to the lack of the corresponding force laws. Some more information can be obtained from chapter 6 where the particles are interconnected by springs leading to equations \({dx^2_i}/dt^2=k(x_{i+1}-x_i) - k(x_i-x_{i-1})\). However, here every considered element interacts with its two neighbours, which does not fit the problem. If one therefore inserts the assumption of footnote 3 (``forces mediating the collision of \(P_{i-1}\) and \(P_i\) cease to act before the forces mediating the collision of \(P_{i}\) and \(P_{i+1}\) begin to act''), then one will be left with \({dx^2_i/dt^2}=k(x_{i+1}-x_i)\). This means that the particles are not interconnected by a spring with spring constant \(k\), but instead are covered with an elastic layer. It still remains, at a first glance, one problem: ``there is one more variable than the number of equations, so that the set of equations has no unique solution''. However, as pointed out in the paper, the set of equations is well defined for a finite set of \(N\) particles: ``The equation of motion governing it would have no term in \(x_{N+1}\)''. In the reviewer's opinion, this fact should also hold in case of infinite number of particles to retain unique definition of collisions (otherwise the model would contradict the predefined behaviour showing ``perfectly elastic collisions from \(P_o\) to \(\dots\)''), thus leading to \[ \begin{pmatrix} \ddot{x}_i \\ \ddot{x}_{i+1}\end{pmatrix}= \begin{pmatrix} +1\cr-1\end{pmatrix} k\Delta x_i;\;\Delta x_i=(x_{i+1}-x_i), \;i=1,\dots, n-1,\;n\rightarrow \infty\tag{a} \] (dot denotes time derivative), in accordance with the ``component convergence'' (2.4): \[ \lim_{ t\rightarrow t^{-}} |V_i(t)-V_i(1) |= 0 \] (\(V\) denotes velocity). The instantaneous collision mentioned in chapter 3 is obtained for \(\Delta x_i=0\), which has then to be considered as a constraint. The corresponding equation of motion reads \[ \begin{pmatrix}\ddot{x}_i\\ \ddot{x}_{i+1}\end{pmatrix}=- \begin{pmatrix} \partial \Delta x_i/\partial x_i \\ \partial \Delta x_i/\partial x_{i+1}\end{pmatrix} \lambda = \begin{pmatrix} +1\\ -1 \end{pmatrix},\lambda \overset{\text{def}}{\Rightarrow} \ddot{\mathbf x}_i = {\mathbf w}\lambda,\tag{b} \] where the spring force converts into the Lagrangian parameter \(\lambda\). Equation (b) can be integrated over the impact interval yielding \( \dot{\mathbf x}_i^+ - \dot{\mathbf x}_i^- = {\mathbf w}\int_{\Delta t_i} \lambda dt \overset{\text{def}}{=} {\mathbf w} \Lambda \enspace \forall t \in \Delta t_i \rightarrow 0. \) Along with Newton's ``impact law'' (which is rather a kinematical constraint than a force law), \( \Delta\dot{\mathbf x}_i^+ + \varepsilon \Delta\dot{\mathbf x}_i^- = (\partial \Delta x_i/\partial {\mathbf x}_i) (\dot{\mathbf x}_i^+ + \varepsilon \dot{\mathbf x}_i^-) = -{\mathbf w}^T(\dot{\mathbf x}_i^+ + \varepsilon\dot{\mathbf x}_i^-) = 0, \) one obtains \[ \dot{\mathbf x}_i^+ = \left[{\mathbf I} - \frac{{\mathbf w} {\mathbf w}^T}{ {\mathbf w}^T {\mathbf w} }(1+\varepsilon) \right]\dot {\mathbf x}_i^- = \frac{1}{2}\left( \begin{matrix} 1-\varepsilon & 1+\varepsilon \\ 1 +\varepsilon& 1-\varepsilon \end{matrix}\right) \dot {\mathbf x}_i^- \] (\({\mathbf I}\) denotes identity, \((\enspace)^T\) denotes ``transposed''). Thus, for \(\varepsilon=0\), and for initial conditions \(\dot{x}_o(t=0)=1\), \(\dot{x}_i(t=0)=0\), \(i=1,\dots, n-1\), \({x}_i(t=0)=\text{const}\), \(i=0,\dots, n-1\) one obtains that the instantaneous velocity switches successively for each particle at the corresponding impact time \(t_i\). From the mechanical point of view, the initially mentioned discontinuity is a result of limiting \(\Delta t_i\rightarrow 0\) and not really a crucial point. The bridge between instantaneous velocity changes and ``smoothing'' is given in the other way by eqs. (a) and (b). Hence, without any risk, one may call the process (b) ``Newtonian'' as long as each collision is defined by Newton's impact law as a limiting case. Clearly, the corresponding succesive sequence of collisions reveals energy and momentum conservation for the whole process, except for the last interaction which cannot be considered as ``usual'' collision since for \(t=1\), according to definition of ST, all masses are at rest. As discussed in chapter 4, the componentwise convergence \(\Delta V_i \rightarrow 0\) will not be sufficient for the whole process if \(V(1)=0\), i.e. the total energy vanishes ad hoc for \(t=1\). In chapter 5, the ST is therefore assumed non-Newtonian, and the conservation of energy is separated from Newton's laws as an independent principle. (It would be more precise to call the process nonconservative since it could still be Newtonian). The ST is considered as an isolated system which does not exchange either matter nor energy with its surroundings (really?) in a mechanical interpretation, the particles 1 to \(n-1\) perform a ``control volume'' where \(P_o\) comes in and \(P_n\) escapes as long as (a) or (b) holds. With definition of ST, \(P_n\) is hindered to leave the control colume, but \(P_o\) is still an input). Energy \(E\) is a step function which is discussed for its approach to \(t=1\) by means of digital representation. It might be somewhat easier to look at the collision events which take place at \(t_k = \sum_{i=0}^{k} (1/2)\cdot(1/2)^n\) (since locations \(x_i\) form a geometrical sequence). Thus, the \((n-1)\)st collision up to where energy conservation holds takes place at \(t^-= 1-(1/2)^n\) which, for sufficient large \(n\), can never be detected numerically nor measured experimentally. (Moreover, the assumption of an independent sequence of collisions does not make sense any more for sufficient high number \(n\)). The authors therefore conclude that \(E(t)\) is undetermined in the neighbourhood of \(t=1\), which leads to a singularity in Newton's equations at this time which is sufficient to refuse to call ST Newtonian. The authors conclude the paper with two points: first, whether or not ST should be called Newtonian, these systems are fascinating and worth a careful study. Second, not all conceivable supertasks constructed with standard Newtonian components are Newtonian even in the weak sense (energy conservation up to \(t^-\)). Indeed, in the reviewer's opinion, there is at least from the mechanical viewpoint something essential missing in the underlying physical model, i.e. a certain construct, or force, which is able to bring the particles at rest for \(t=1\). Then, given the ST as definition wherever it may come from, any non-Newtonian, or artificial, process might be considered as its reason. There remains one question: is ST assumed a physical process, or are we leaving here physics at all? The time history of predefined ST (perfect elastic collisions between the partices but staying at rest at the end of maneuver) is, from the mechanical view point, strange.
scientific article; zbMATH DE number 1563711