Higher-Order Dynamics

Given an isolated set of n non-interacting particles of masses m₁, m₂, …, m_n, and velocities (confined to one spatial dimension for simplicity) of v₁, v₂, …, v_n, we can define an infinite sequence of “parameters of motion”

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 non-interacting, the law of inertia implies that each of the masses and velocities are individually constant, and hence all the higher parameters of motion, c₃, c₄, …, are also constant. We will assume here, as in Newtonian mechanics, that each of the individual masses is constant, so c₀ 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 c₁ and c₂ are to be held constant. This gives us two equations in the two unknown velocities v₁ and v₂

These equations imply that v₁ and v₂ are the (matched) roots of the quadratic equations

where (recall) c₀ = m₁ + m₂, so there are two discrete solutions for (v₁,v₂), namely

which shows that

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

The parameters of motion for the two interacting particles c_j are related by the second-order recurrence

where s₁ = v₁ + v₂ and s₂ = v₁v₂ are the elementary symmetric polynomials of the velocities, from which we can compute v₁ and v₂ as the roots of v² – s₁v + s₂ = 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

Thus the two distinct velocity solutions consistent with a given c₁ and c₂ correspond to the two distinct values of c₃ (as well as all higher constants of motion c₄, c₅, … etc.) As a result, only the first two “parameters of motion” are conserved through binary interactions. Obviously c₁ is the momentum of the two particles and c₂ 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 v₁, v₂, v₃ of the three particles, we need three equations, and by analogy with the binary interactions for which c₁ and c₂ are conserved (along with c₀), it’s natural to suppose that c₁, c₂, and c₃ are conserved in trinary interactions. These conditions are of degree 1, 2, and 3 respectively, so we expect 3! = 6 discrete solutions (v₁,v₂,v₃) corresponding to any given values of c₁, c₂, and c₃. We can determine the sixth-degree 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 c₁ = 0. Then the ith velocity is a root of the polynomial

where

and the quantities σ₁ and σ₂ are the elementary symmetric polynomials of the “other” two masses divided by the ith mass. For example, the coefficients of the polynomial for v₁ are defined in terms of the symmetric functions

It’s worth noting that A₅ = 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 sixth-degree polynomial, note that if m₁ = m₂ = m₃ we have σ₁ = 2 and σ₂ = 1 and the polynomial is

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:

The six possible states of this system are

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 third-order recurrence

where

are the elementary symmetric polynomials of the velocities, from which we can compute v₁ and v₂ as the roots of v³ – s₁v² + s₂v – s₃ = 0. Therefore, given the first five parameters of motion, we can determine the elementary symmetric polynomials of the velocities by solving the system

Thus the six distinct solutions consistent with given values of c₁, c₂, and c₃ correspond to six possible sets of values of the parameters c₄ and c₅, as well as of all the higher-order 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 c₀, c₁, and c₂, but of c₃ 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 higher-order 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 lower-order interactions. We also get strict conservation of all the parameters of motion from c₀ 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 higher-order 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 higher-order 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 co-located 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 point-like, then they would never interact, whereas if they possess some sharp non-zero 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 higher-order interactions.

Even if we insist on interpreting the interacting entities as point-like particles, the non-positive-definite nature of the Minkowski spacetime metric still suggests the possibility of higher-order interactions. In Galilean spacetime the spatio-temporal 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 higher-order interactions need not violate the principle of locality.

Whether a system of infinitely many particles could be treated in terms of infinite-order 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 non-conserved parameters (like the parameters c₄ and c₅ 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.

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