Accumulating or constraining

Mm11 WebTitle{Accumulating or constraining}

Processes (evolutions) or situations (states)

For many processes studied in classical physics we think we've acquired some understanding of the process when we can write down a fixed deterministic rule that describes how the process evolves: what future state (there is only one — that's determinism) follows from the current state.

The paradigm for much of this aspiration is the dynamical system of Newtonian mechanics. The rule of evolution for this dynamical system is a set of relationships that link the state now to the state a short time in the future, given as differential or difference equations. To determine the states for all future times just keep iterating the relation many times: advancing step by step by step.

Processes evolve through accumulations.

There are a few relationships like this in elementary physics, but only a few. Many relationships do not specify how a state will evolve into another: they are not concerned with specifying how a situation evolves over time — a process — rather they are concerned with relationships between different quantities in a given situation. These situations are more formally referred to as states.

A state quantity or state variable is a property of a situation that depends only on what's the case now, not on how that situation was reached, or even where it will go next. In thermodynamics a state variable describes the equilibrium state of a system — for example, thermal energy. These are atemporal relationships, as there is no evolution over time: indeed no first one, then another.

Relationships that predict changes in quantities over processes are accumulations(quantities with accumulations: QWA): those that relate quantities in situations are constraints(quantities with constraints: QWC).

Fundamentally different connections between quantities


Accumulation is all about fluxions: about changes over time. Newton was deeply interested in change over time. His invention of the calculus enabled a more precise description of how quantities changed over time: how they accumulated in either a positive or negative sense. The accumulating quantity Newton called a fluent, the quantity which set the accumulations a fluxion. These quantities were central to his publication: "The Method of Fluxions".

To enable children to work with changes over time I think we might well gain from a return to this concern of tracking the change of a single quality through time. We could imagine mapping this onto a fluent-space, or fSpace. This would be similar to a sparkline, popularised by Tufte, as a common way of tracking quantities over time.

Graphs and Accumulations

Accumulations are all about change, not about the accumulated effect, so there is a focus in CMR on watching representations of quantities evolve over time, rather looked back on a process that has happened.

Conventional scatter graphs are frozen rivers, perhaps recording what happened to the fluents and the fluxions, and invite geometrical rather than fluxion-based reasoning.


Constraints are all about possibility spaces. In a situation, or state, it's often the case that discoveries have been made about the values that different quantities can have: not every quantity can have any values—there are constraints. There are two facets to establish in introducing the discovery:

Representing these relationships with a scatter graph is a geometrisation of 'If this then that'(IFTT), or a "game of consequences": the constraint relationships define the possibility space. Here I'll prefer the phrase "possibility space" to "consequence space" as it reduces any tendency to think of time flowing as the consequences evolve (similarly, I'd go light on comparison with IFTT as this is (currently) a temporal, causal internet service). It is the case that these kinds of relationship are most common in introductory physics, and least often explicitly modelled, perhaps because nothing happens by itself, and it is quite hard computationally.

This could be a very useful precursor to thinking about energy as this also defines possibility spaces (and impossibility spaces, by analogy to the art world's figure and ground—you just cannot go there), although these are more extensive. It's maybe an introduction to phase space, as well.

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