HigherOrder Dynamics 

Given an isolated set of n noninteracting particles of masses m_{1}, m_{2}, …, m_{n}, and velocities (confined to one spatial dimension for simplicity) of v_{1}, v_{2}, …, v_{n}, we can define an infinite sequence of “parameters of motion” 

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The first of these represents the conservation of total mass, the second represents conservation of total momentum, and the third represents conservation of total kinetic energy. Of course, since we’ve assumed the particles are noninteracting, the law of inertia implies that each of the masses and velocities are individually constant, and hence all the higher parameters of motion, c_{3}, c_{4}, …, are also constant. We will assume here, as in Newtonian mechanics, that each of the individual masses is constant, so c_{0} is automatically conserved. 

Now we consider the possibility of “interactions” between the particles. Normally we restrict our attention to interactions involving just two particles, so we need some principle(s) to constrain the two continuous degrees of freedom represented by the velocities of two given particles. For this purpose we stipulate that c_{1} and c_{2} are to be held constant. This gives us two equations in the two unknown velocities v_{1} and v_{2} 

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These equations imply that v_{1} and v_{2} are the (matched) roots of the quadratic equations 

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so there are two discrete solutions for (v_{1},v_{2}), namely 

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which shows that 

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Corresponding to the two discrete solutions are the two possible values of the third parameter of motion 

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The parameters of motion for the two interacting particles c_{j} are related by the secondorder recurrence 

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where s_{1} = v_{1} + v_{2} and s_{2} = v_{1}v_{2} are the elementary symmetric polynomials of the velocities, from which we can compute v_{1} and v_{2} as the roots of v^{2} – s_{1}v + s_{2} = 0. Conversely, given the first three of these parameters of motion, we can determine the elementary symmetric polynomials of the velocities by solving the system 

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Thus the two distinct velocity solutions consistent with a given c_{1} and c_{2} correspond to the two distinct values of c_{3} (as well as all higher constants of motion c_{4}, c_{5}, … etc.) As a result, only the first two “parameters of motion” are conserved through binary interactions. Obviously c_{1} is the momentum of the two particles and c_{2} is twice the kinetic energy, which explains why these two parameters are so important in the dynamics of binary interactions. Moreover, if we assume that all interactions can be decomposed into elementary binary interactions, it follows that momentum and kinetic energy (along with total mass) are sufficient to characterize the dynamics of all systems of particles and their (conservative) interactions. 

However, the assumption that all interactions can be reduced to binary interactions is somewhat arbitrary. For example, it’s possible to conceive of trinary interactions, i.e., primitive interactions between three particles, not reducible to binary interactions. To constrain all three of the continuous degrees of freedom represented by the velocities v_{1}, v_{2}, v_{3} of the three particles, we need three equations, and by analogy with the binary interactions for which c_{1} and c_{2} are conserved (along with c_{0}), it’s natural to suppose that c_{1}, c_{2}, and c_{3} are conserved in trinary interactions. These conditions are of degree 1, 2, and 3 respectively, so we expect 3! = 6 discrete solutions (v_{1},v_{2},v_{3}) corresponding to any given values of c_{1}, c_{2}, and c_{3}. We can determine the sixthdegree polynomial for each velocity by eliminating the other two velocities from the governing equations. It’s most convenient to work in the rest frame of the center of mass, so that c_{1} = 0. Then the ith velocity is a root of the polynomial 

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where 
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and the quantities s_{1} and s_{2} are the elementary symmetric polynomials of the “other” two masses divided by the ith mass. For example, the coefficients of the polynomial for v_{1} are defined in terms of the symmetric functions 

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It’s worth noting that A_{5} = 0, signifying that the sum of the six possible values for the velocity of any given particle is zero (with respect to the rest frame of the system’s center of mass). As a check on the correctness of the sixthdegree polynomial, note that if m_{1} = m_{2} = m_{3} we have s_{1} = 2 and s_{2} = 1 and the polynomial is 

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This confirms that, with equal masses, the six possible states are just the six permutations of three velocities (which of course sum to zero). On the other hand, if the masses are not equal, we generally have six distinct values for each of the velocities. As an example, consider a system of three particles (in one space and one time dimension) with the following masses and parameters of motion: 

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The six possible states of this system are 

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As mentioned, the sum of each column is zero. Interestingly, for each state there is another state such that one of the velocities is roughly the same and the other two are transposed. In other words, to some extent these particular trinary interactions mimic binary interactions. But there are also transitions for which all three of the velocities change significantly. Also, none of these exact transitions is achievable by any combination of successive binary interactions. 

The parameters of motion c_{j} for three interacting particles are related by the thirdorder recurrence 

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where 
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are the elementary symmetric polynomials of the velocities, from which we can compute v_{1} and v_{2} as the roots of v^{3} – s_{1}v^{2} + s_{2}v – s_{3} = 0. Therefore, given the first five parameters of motion, we can determine the elementary symmetric polynomials of the velocities by solving the system 

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Thus the six distinct solutions consistent with given values of c_{1}, c_{2}, and c_{3} correspond to six possible sets of values of the parameters c_{4} and c_{5}, as well as of all the higherorder parameters of motion. 

The existence of six possible states, instead of just two, might be regarded as a loss of predictive strength, but in compensation we now have strict conservation not just of c_{0}, c_{1}, and c_{2}, but of c_{3} as well. Furthermore, each interaction involves – to some degree – every particle in the system, reminiscent of Mach’s principle. One of the inherent shortcomings of most scientific theories is that they must consider “isolated systems”, even though ultimately the isolation is untenable. Proceeding to even higherorder interactions, we find that the nth order interactions among n particles allow n! possible states, but again we anticipate that many pairs of these states are related by transitions that mimic lowerorder interactions. We also get strict conservation of all the parameters of motion from c_{0} to c_{n}. It’s interesting to consider the limit of this process, in which we consider all particles (which may be a huge finite number, or infinitely many). The set of possible states is always finite and the states are always discrete for any finite n, but it becomes extremely large. We would then have strict conservation of all meaningful parameters of motion, and the entire universe would be, in effect, a single state. Evidently some other principle(s) would then be needed to select the actual state, or we could imagine a superposition of all the n! possible states, with some weight factors expressing the selection principles. 

It’s interesting to consider the overall contrast between purely binary dynamics and higherorder dynamics. When viewed in terms of purely binary interactions, a given set of parameters of motion (for two particles) can be inferred from the current state of those two particles, and it then uniquely determines the single other possible state consistent with those parameters. Thus the future state of these two particles can be uniquely inferred from their current state. The “flow of implication” is traced by considering first one pair of particles (A,B) transitioning to the alternate state (A’,B’), and then considering the pair (B’,C), which transitions to (B”,C’), and so on. At each stage the two local parameters of motion (momentum and energy) are conserved, and the interactions are assumed to occur when the particles coincide, thereby establishing the ordering of events. With higherorder dynamics we could, in principle, determine the large number of parameters of motion for any given state, but then we are confronted with a huge number of possible alternate states characterized by those same parameters – despite the fact that there are no continuous degrees of freedom. This has an agreeable holistic quality, seeming to avoid the “measurement problem”, and all possible states of the overall system are already explicitly delineated in the set of solutions. But there is no obvious “ordering” or sequencing of these states. This somewhat resembles the evolution of the wave function of a system in quantum mechanics. A system doesn’t evolve through a sequence of specific observable states, but rather into a superposition of all possible states. Still, we arrive back at the measurement problem when trying to decide how a measurement yields (or seems to yield) just one of the observable states. 

The usual justification for considering only binary interactions, and also for imagining that subsets of all the particles can be considered as “isolated systems”, is that interactions occur only between colocated particles. However, this is a problematic notion at best, since it relies on a specific and not entirely intelligible model of particles and their interactions. If particles are considered to be of sharply defined extent and pointlike, then they would never interact, whereas if they possess some sharp nonzero extension they possess separate parts with some forces required to maintain their sizes and shapes, so they can hardly be considered elemental. More consistent with modern concepts of “particles” is the idea that they are manifestations of fields, and as such they are of infinite extent, all overlapping with each other. From this point of view it’s not unnatural to consider higherorder interactions. 

Even if we insist on interpreting the interacting entities as pointlike particles, the nonpositivedefinite nature of the Minkowski spacetime metric still suggests the possibility of higherorder interactions. In Galilean spacetime the spatiotemporal separation between two particles is zero only if those particles coincide, whereas in Minkowski spacetime every two particles are connected by infinitely many null intervals along their worldlines, even if the worldlines never intersect each other. In fact, multiple separate particles can all lie along a single null interval, so higherorder interactions need not violate the principle of locality. 

Whether a system of infinitely many particles could be treated in terms of infiniteorder interactions is unclear. On one hand this would seem to allow us to assert the conservation of all the parameters of motion c_{j} for j = 0 to infinity, but on the other hand the matrix that determines the velocities must involve infinitely many nonconserved parameters (like the parameters c_{4} and c_{5} in the case n = 3). From a purely mathematical standpoint, it would be interesting to know whether going to the limit of infinitely many particles would still yield discrete countable states, or whether it would result in an uncountable continuum of possible states. 
