#configuration space

[isomorphismes]

Amie Wilkinson

the configuration space of two unmarked (same color) points on the circle is a Möbius band, which is covered by the cylinder, the configuration space of two marked (differently colored) points

[endfall]

If you overlay every julia set on top of every other julia, this is what you get... sort of. The precise technique is a bit more nuanced, and I’ll write a post on it eventually. Still, it’s it neat that even this “configuration space of all julia sets” shows such structure. As for why it’s symetric with respect to the origin, think about how Julia sets change as you go over the complex plane. (If you’ve never done this, the fractal program Xaos has a mode called “Fast Julia” that lets you see it in real time.)

[isomorphismes]

If homology classes in configuration spaces were viewed as planetary systems, cohomology classes would represent planetary alignments.

Dev P. Sinha, Homology of the little disks operad

[gynoidwren]

Language Space

This is an idea I had after reading Julian Barbour's book The End Of Time. I call it Language Space.

Language Space is a structure through which we can use physics to visualize and understand languages and how they evolve. It comes from the need for a way to objectively define language and dialect, and is driven by the understanding that fundamentally we each speak our own "language." Each of us has unique habits that fit within a realm of variations: personal speech patterns within a dialect within a language within a linguistic family, a kind of linguistic taxonomy. We can thus define a mathematical coordinate space in which each person's unique usage is represented by a point within this space. A point of matter. In watching these points collect within Language Space we will see areas of coalescence, both on a small and a grander scale, like solar systems and galaxies. These areas represent relations between different speakers' linguistic habits.

We can thus mathematically determine the centres of these areas and objectively define that as the "root" of the language. We can take everything within a certain radius (based on the distribution of "matter") of that point as the language itself, ultimately making each language a kind of galaxy and each language group a galactic cluster within our new metaphorical universe in Language Space. We can use the same process for defining dialects as with languages, but on a smaller scale, finding the centres of areas of high density and using the radial distance out to the area where that mass truncates (or, more realistically, falls off significantly) to define the area of the dialect.

Language Space would also lend some objectivity to the ambiguously-defined notion of a dialect, wherein currently the same term is used to refer to the distinction between Mandarin from Cantonese as is used to distinguish between British and American English, or indeed New England and California English. Perhaps this would necessitate the creation of a new term (or several), but it would allow us to note that the differences between the above distinctions are ones involving objective radial distances from the language centre and quantitative scale differences in the distribution of linguistic "matter." Not only can we define dialect objectively, but we can compare the "sizes" (or scales of variation), rates of change, and relationships of different dialects and other trends in language.

This is all well and good, but here is where the fun really comes in: just as the real universe changes, we can use "movement" through Language Space to model how a language or dialect evolves over time. This provides great potential, and it is also the point from which I would provide my most controversial insight: that the laws of physics apply to languages. We must then ask what these concepts mean in this context. Gravity is certainly both the most basic and the most obvious: our differing usages of language tend towards harmony as we have the dual tendencies both to correct each other and to adopt others' usage paradigms, either through learning or unconscious imitation. This gives rise to the interesting probability (and a seemingly likely one, at that) of an eventual "Big Crunch" in Language Space similar to the possible one in our universe: that is the collapse of all language groups, probably not quite into a singularity as in our universe, but into a single monolithic language. A minorly-fluctuating steady state.

Other concepts seem to apply as well, but their implications are more unclear. It might be interesting to ask whether this is also an expanding universe, like our own, wherein the increasing possibilities are causing language groups to grow further apart. I doubt this very much, primarily due to the way that Language Space itself is constructed, but it's an interesting possibility. Inertia, and consequently momentum, might be seen in the developmental process both as children learn a language for the first time and as people revise their own language use. Thus, those who are already making adjustments, learning a new language, etc. are more likely to continue making revisions to their own personal usage while people obviously have a tendency to remain unchanging when not so due to their comfort in the way they have learned language and due to their own usage of language defining their own reality. The former case would give interesting implications about those learning a second (or third, etc.) language.

Velocity is also useful, and related to the previous idea of momentum. Trends in changing language use tend to continue, and the strength which they do so could be a measurement function of the rate at which they move in Language Space. Or, put differently, just as velocity is the derivative of position, a rate of change, the derivative of position in Language Space would similarly yield the rate at which a language, dialect, or indeed a personal use pattern is changing. This could lead to a notion of orbits. Perhaps some dialects orbit their language centres, while others are falling in. What happens when two dialects collide and combine? This intertwining process could be quantitatively measured and mapped out in Language Space.

Of course, the problem with the practicality of this implementation is that it is naturally impossible for us to visualize, or indeed hope to define, Language Space quite as described. We are, after all, talking about what is essentially a multidimensional configuration space for the possibilities of linguistic communication. Indeed, if we were to use as a coordinate every variable that would influence or define one's personal use of language and/or how it changes, language space would have a very large (near-infinite, in fact, perhaps a child's notion of infinity) number of dimensions to it. Even if we restrict all such dimensions to a certain class, as an example the inclusion of a single word in someone's vocabulary, the result would be mind-bogglingly complex. Language space is then, by necessity, something to be conceived of mathematically and not visually, if not even entirely in the abstract.

However, if this hurdle can be leapt over, the mathematics do exist. Configuration spaces are well understood, even if they are not often (or ever, basically) implemented for such complex systems. The results seem clear, though, that if this is an obstacle that it is possible to surmount, the possible uses and boons of Language Space are immense. In fact, even as an abstract concept it can tell us things, as outlined above. Further study and elaboration is necessary.

[isyeye]

Configuration space

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, which means that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

Configuration spaces in physics

The configuration space of a single particle moving in ordinary Euclidean 3-space is just R3. For n particles the configuration space isR3n, or possibly the subspace where no two positions are equal. More generally, one can regard the configuration space of n particles moving in a manifold M as the function space Mn.

To take account of both position and momenta one moves to the cotangent bundle of the configuration manifold. This larger manifold is called the phase space of the system. In short, a configuration space is typically "half" of (see Lagrangian distribution) a phase spacethat is constructed from a function space.

In quantum mechanics one formulation emphasises 'histories' as configurations.

[isyeye]

Configuration Space

In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system. It is often the case that these parameters satisfy mathematical constraints, which means that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold is called the configuration manifold of the system.

Configuration spaces in physics

The configuration space of a single particle moving in ordinary Euclidean 3-space is just R3. For n particles the configuration space is R3n, or possibly the subspace where no two positions are equal. More generally, one can regard the configuration space of n particles moving in a manifold M as the function space Mn.

To take account of both position and momenta one moves to the cotangent bundle of the configuration manifold. This larger manifold is called the phase space of the system. In short, a configuration space is typically "half" of (see Lagrangian distribution) a phase space that is constructed from a function space.

In quantum mechanics one formulation emphasises 'histories' as configurations.

[designprobz]

The grid endures in Configuration Space, an App for iPad that superimposes a 3D Cartesian grid over the perceptual field. Configuration Space was developed by artist Jesse Harding.

[sayitwithscience]

Originally, in the Newtonian formulation of classical mechanics, equations of motion were determined by summing up vector forces (à la free body diagrams). Is there a different way to find the equations of motion?

In place of drawing a free body diagram, we can represent a system more rigorously by describing its configuration space. The configuration space (often denoted Q) of a system is a mathematical space (a differential manifold) where every point in the space represents a particular state or configuration of the system. A curve drawn through a configuration space, then, represents the evolution of a system through a sequence of configurations.

Consider a rod along which a pendulum can slide. We need two numbers to describe the state of this system: the angle of the swinging pendulum and the position of the pendulum’s base along the rod. These two numbers are generalized coordinates for our system. Just like a traditional, linear vector space has a coordinate basis (like x, y, and z), our configuration space can use our generalized coordinates as a basis; let’s choose to name the position on the rod x and the angle of the pendulum φ. Since x can take any real value and φ can take any value from 0 to 2π (or 0 to 360o, if you like), the x dimension can be represented by a line (R1) and the φ dimension by a circle (or a one-sphere, S1). When we combine these dimensions, our new space — the configuration space of this system — is shaped like an infinite cylinder, R1 x S1. Just imagine connecting a circle to every point on a line… or, conversely, a line to every point on a circle.

The general process of examining a system and the constraints on its movement is a standard first step for solving mechanics problems analytically. After accounting for the constraints on a system, the ways a system can vary are called the degrees of freedom. Their count is often represented by the variable s. Notice: s = dim(Q).

Now that we’ve represented the configuration of our system, we need to talk about the forces present. There are several different ways that we can set up scalar fields on our configuration manifold that represent quantities related to the energy of the system. The simplest to deal with is often the Lagrangian, L = T - V = (Total kinetic energy) - (Total potential energy). Some fancy mathematics (a.k.a. calculus of variations) shows that when we define the Lagrangian in terms of our coordinates and their time derivatives, we can easily derive the equations of motion using the Euler-Lagrange equation.

For more complicated systems, configurations spaces may look different. A double pendulum (a pendulum on a pendulum) would have the topology S1 x S1 = T2, the torus (as pictured). Many systems will have higher dimensions that prevent them from being easily visualized.

Exercise left to the reader: the Lagrangian explicitly takes the time derivatives of the coordinates as arguments; information about the velocities of the system is needed to derive the equations of motion. But this information isn’t included in Q, so Lagrangian dynamics actually happens on TQ, the tangent bundle to Q. This new manifold includes information about how the system changes from every given configuration; since it needs to include a velocity coordinate for each configuration coordinate, dim(TQ) = 2s. TQ is also called Γv, the velocity phase space. T*Q, the cotangent bundle to Q, is the dual of TQ, and is traditionally just called the phase space, Γ; this is where Hamiltonian mechanics takes place.