Rigidity in Curved Space

This is a short discussion of what it means to model a physical object in curved space, to help me sort out the correct definitions for an upcoming paper on simulations with Brian Day and Sabetta Matsumoto.

Modeling

We separate out the problem of (theoretically) modeling a physical system into two steps: statics and dynamics. We also divide the problem of (practically) modeling a physical system into steps: the construction of a numerical scheme to simulate the evolution of the system, an implementation of this scheme, and means of verification that our implementation correctly simulates the dynamics.

Statics: The problem of statics is to determine what states an object can find itself in, and what kind of mathematical space is suitable for modeling it. This problem is rather subtle for general deformable objects (e.g. are topology changes allowed?), but for the cases of present interest configuation spaces such as those familiar from introductory physics often suffice.

Dynamics: Given a space of states for an object, the question of dynamics is to find the correct evolutionary equation on this domain whose solutions track the time evolution of the given system under the laws of classical physics. Within the general framework of Lagrangian (resp. Hamiltonian) mechanics, this evolution equation will take the form of a vector field on the tangent (resp. cotangent) bundle to state space, whose integral curves trace the physical evolution of the system from the starting intial condition (co)tangent vector.

Numerical Schemes: After making explicit the theory, the main body of this work involves simulation, or the development of numerical schemes to approximate these evolutionary equations Given an evolutionary equation defined on some abstract space of states, While some mathematical subtleties already are present in many introductory physics problems (computing in generalized coordinates on a configuration space, coarse-graining an infinite dimensional problem to find a suitable finite-dimensional analog, among others), a slew of new difficulties arise when the background space(time) of the problem has nonzero curvature. Indeed, with nonzero curvature we must take seriously the difference between points and tangent vectors, rendering even simple numerical update schemes such as $x_{n+1}=x_n+v_n\Delta t$ nonsensical. Properly dealing with these issues is one of the main contributions of this work.

Validation: Finally, given a numerical scheme, how can we evaluate its accuracy and stability? Provided the appropriate theoretical work can be done, some natural metrics involve comparison with analytical solutions, and the tracking of conserved quantities respectively.

What is an Object?

Leaving aside the philosophical, ship-of-Theseusesque concerns over the conceptual coherence the of ‘objects’ more generally, accurately modeling even the statics of deformable bodies in a curved background space presents many practical difficulties. To give a taste of the issues that need to be considered, here are several questions that must be answered before fixing some model of the space of states:

Topological: How much is an object allowed to change shape before it becomes a different object? An inflated and deflated balloon are certainly the same object, but how about after it has been popped? Are rain droplets individual objects? What happens when they strike the surface of a pond and merge?

Isotopic: Are an object and its mirror image two states of the same object, or different objects entirely? More generally, what should we say about two things that are topolgoically and geometrically identical, but there is no way to continuously move one to the other? While mirrors proivde the main example of interst in flat space, this consideration becomes much more involved in curved space, especially for rigid bodies.

Diffeomorphic: Does the precise layout of points inside an object matter, or is it only its total shape as a subset of the ambient space that is relevant? Mathematically, we need to know if applying a diffeomorphism that fixes an object setwise counts as ‘doing something’. A physical situation where this is relevant is the distinction between an object made out of rubber (where internal diffeomorphisms are important and affect the internal physics, contributing to stress and strain) and an object filled with air (where we may wish to treat the interior as a homogeneous substance and abstract away any turbulence / internal flows from our model).

Physical: A physical object is more than just the location of its points in space: how do we model properties like mass and internal fores? If two things have the same shape but different mass densities, are they two different objects or two versions of the same object? Can the potential energy change independent of the physical shape/location of the object, or does that make it a different object (perhaps having different internal structure, and thus different internal forces)?

Geometry of the Ambient Space

The properties of the background space $X$ relevant to classical physics include things like relative distances, velocities, angles and speeds. All of these are derivable from the mathematical data of a choice of Riemannian metric $g$ on $X$. Thus, we allow as ambient space in our model a choice of arbitrary smooth manifold $X$ and Riemannian metric $g$ on $X$. Upon fixing such a $g$, we will refer to the entire Riemannian manifold $(X,g)$ simply as $X$.

Statics of Deformable Objects

External Configurations: We restrict ourselves here to the case that deformable objects cannot undergo topological chagnes while remaining ’the same object’. For specificity, we may fix this topological type by choosing a topological space $B$, where all versions of our object must be homeomorphic to $B$. We will call this space $B$ the reference object.

To a first approximation (ignoring internal states), one may wish to think of an object of topological type $B$ in the ambient geometry $X$ as a subset of $X$ which is homeomorphic to $B$. To account for internal states of the object, we must track more data than simply a subset of $X$. The natural generalization is to record an object in $X$ as an embedding $q\colon B\to X$, where we think of $q$ as giving a rule for exactly how to place the reference object into the ambient space. Because of constraints, we often do not wish to say that every such embedding is a possible state of the object, but rather that our space $Q$ of all states is a subset of the space $\mathrm{Emb}(B,X)$ of embeddings. A useful general calss of such constraints (essential in dealing with rigidity) are holonomic constraints, where we take $Q$ to be a closed submanifold of $\mathrm{Emb}(B,X)$.

Internal Physics: The model so far accounts for the objects position (and geometric constraints on its deformation), but does not yet incorporate and internal parameters relevant to the physics. To incorporate the mass of our object, we specify a measure, which we will call the mass density $dm$, on the reference object $B$. This measure stores the information about the internal density of our object; given an embedding $q\in Q$ the density of the physical object $q(B)\subset X$ is given by the pushforward measure $q_\star dm$ on $q(X).

The other internal physical characteristic of interest is the internal forces of an object. When such forces completely dominate the internal dynamics (such as being strong enough to enforce absolute rigidity), they may already be accounted for by choosing constraints on the space $Q$ of configurations above. More general internal forces, which affect but do not dominate the dynamics, are modeled by specifying a potential energy $V$. This potential energy is a function of the shape / location of our object in space, and thus determines a real valued function $V\colon Q\to \mathbb{R}$ (which may also depend on the aforementioned mass density $dm$).

General Definition: Putting both of these components together allows for a rather general model for working with deformable objects in curved space. A deformable object in a Riemannian manifold $X$ is a closed submanifold $Q$ (possibly the entire thing) of the space of embeddings of a given reference object $B$ into a geometry $X$, together with a measure (the mass density) $dm$ on $B$ and a real valued function $V$ (the potential energy) on $Q$. Formally, we track a deformable object by the 4-tuple $(B, dm, Q, V)$; though we will refer to the object simply as $B$ when the additional data is understood.

Dynamics of Deformable Objects

Such a system is very amenable to the standard treatment of classical physics, specifically in the Lagrangian or Hamiltonian formalism. To model free motion in the ambient space $X$, the only additional data required is a norm $K$ on the tangent bundle to $Q$ measuring the kinetic energy. The Riemannian metric $g$ on the ambient space $X$ and the mass density $dm$ determine $K$ uniquely: if $q\colon B\to X$ is an instance of our object, and $v\colon B\to TX$ is a tangent vector to $Q$ at $\phi$, then $$K(v)=\frac{1}{2}\int_{x\in B} g_{q(x)}(v(x),v(x))dm$$ When $B$ is a finite point set and $dm$ is a sum of dirac measures giving the mass of each point, this integral reduces to the familiar $K(v)=\frac{1}{2}\sum_i m_i g_{q_i}(v_i,v_i)$, where $m_i$ is the mass of the $i^{th}$ point, $v_i$ is its velocity and $q_i$ is its location.

Together with the potential energy $V$, this gives the Lagrangian $\mathcal{L}$ for $B$, as a function $TQ\to \mathbb{R}$:

$$\mathcal{L}(q,v)=K(q,v)-V(q)$$

From this lagrangian, we may write down the associated quations of motion in a choice of generalized coordiantes in one of two ways. First, if either 1) $Q$ is the entire space of embeddings of $B$ into $X$ or 2) we construct a new set of generalized coordinates on $Q$ directly, then evolution is given directly by the unconstrained Euler-Lagrange equations for $\mathcal{L}$, where $q^i$ is a coordinate on $Q$ and $v^i$ is the coordinate in direction $\partial q^i$ on $TQ$.

$$\frac{\partial \mathcal{L}}{\partial q^i}-\frac{\mathrm{d}}{\mathrm{dt}}\frac{\partial\mathcal{L}}{\partial v^i}=0$$

Alternatively, we may wish to use coordiantes $q^i$ on the entire space of embeddings $B\to X$, even when the state space of our system is a proper submanifold (for example, if we wish to constrain the volume of the object, or enforce rigidtiy). In this case, we avoid the construction of a suitable set of generalized coordinates by paying the price of using the constrained Euler Lagrange equations for $\mathcal{L}$. Let ${f_j}$ be a set of constraints defining $Q$: that is, a set of real valued functions on $\mathrm{Emb}(B,X)$ cutting out $Q$ as a level set (for fixed $x_j\in\mathbb{R}$), $Q={q\in\mathrm{Emb}(B,X)\mid f_j(q)=x_j}$. Then the equations of motion are given as follows

$$\frac{\partial \mathcal{L}}{\partial q^i}-\frac{\mathrm{d}}{\mathrm{dt}}\frac{\partial\mathcal{L}}{\partial v^i}=\sum_j \lambda_j\frac{\partial f_j}{\partial q^i}$$

Together these two evolutionary equations completely define the dynamics of general deformable objects (so long as the constraints are holonomic).

Key Examples:

The definition above is very general, and encompassing both discrete and continuous systems, constrained and unconstrained systems, systems with variable or constant mass, with rigid and nonrigid components, etc. This zoo of examples can be constructed by varying the choice of constraint system and potential energy respectively, and to exhibit the breadth of possible phenomena, we will single out special cases highlighting each of these separately.

Rigid Objects

Rigid objects are defined by the property that the relative distance between any two points of the object remains unchanged under time evolution. In our formalism, this amounts to a system with constraints but no potential. These systems are relatively straightforward individually, but provide a tractable playground for understanding the interaction of objects in curved spaces.

Statics: The fact that rigid objects cannot have their internal components move relative to eachother naturally defines a constraint on the possible embeddings in $\mathrm{Emb}(B,X)$ which may represent states of the object Given one admissable embedding $q\in \mathrm{Emb}(B,X)$, if $q^\prime$ is any other admissable embedding, the composition $q^\prime q^{-1}$ is an isometry from $q(X)$ to $q^\prime(X)$. Because the metric information of the object is constant across embeddings, in the rigid case we may think of this as data about the reference object, endowing $B$ itself with a distance function $d\colon B\times B\to \mathbb{R}$. From this perspective we can specify $Q$ as the subset of $\mathrm{Emb}(B,X)$ which are isometric embeddings.

When $X$ is homogeneous, this set $Q$ of embeddings naturally identifies with the isometry group of $X$: given any isometric embedding of our reference object $B$, every other embedding of $B$ is simply this embedding followed by some global isometry. Thus, for the study of rigid body dynamics, the Lie group $G$ of symmetries of $X$ plays a fundamental role.

To complete our description of a rigid object, we give a mass density $dm$; but we do not need to specify a potential energy (equivalently, we may think of the potential energy as just equal to zero for all configurations in $Q$). All information about the internal dynamics and relative motion between components of the object have been completely specified by the imposition of complete rigidity, so there is no need nor room for the consideration of internal forces beyond this.

Dynamics: The configuration space of the system is constrained, so we must either 1) parameterize the submanifold $Q$ of admissable configurations to do Lagrangian mechanics directly on $Q$, or 2) work with the constrained Euler Lagrange equations. In general 2) may be the only available option, but in the case of homogeneous ambient geometries, the fact that $Q$ naturally identifies with the Lie group $G$ of global isometries of $X$ gives a natural means of implementing 1).

For a single rigid object then, the kinetic energy determines a Riemannian metric on the isometry group of the ambient space, and the geodesics of this metric trace out the time evolution of various initial conditions for this rigid body. (In the familiar case of Euclidean space, the geodesics of the Euclidean group correspond to geodesic motion for the center of mass of a body, while rotating about some axis).

Multiple Objects: The physics of a single rigid body in a homogeneous space straightforwardly reduces to the computation of geodesics for a metric on the isometry group $G$, and while this may still take some work to compute for complex objects, the true interesting behavior for rigid objects lies in their interaction. After choosing a reference embedding, the state space of any individual rigid object is parameterized by the group $G$, and so the total state space for a collection of finitely many objects naturally identifies with a subset of the product of finitely many copies of $G$. (It is a subset, and not the whole space, as the objects take up physical space and we do not wish to allow overlaps) This complicates the analysis as the state space is cut out of $G\times G\times\cdots\times G$ by a system of non-holonomic constraints. To understand the dynamics at the points of interaction, it is useful to consider the conserved quantities of the unconstrained system. DO THIS! STILL NEED TO ‘DERIVE’ THE COLLISION ALGORITHM…

Squishy Objects

On the other side of the deformable object spectrum, we consider squishy objects to be those whose internal physics are dominated by local internal forces, as opposed to global constraints. These objects take seriously the microphysical nature of matter and model the opposition to changes in shape through a conspiracy of internal forces, and

Statics: Squishy objects allow for deformations in shape, and so the state space is unconstrained, and can be taken to be the entire space of embeddings $\mathrm{Emb}(B,X)$). The internal physics of a squishy object is not encoded by an enforced constraint on the space of embeddings, but rather by specifying a potential energy on the space of states $Q$. An common choice of potential is the harmonic potential, which enforces a spring-like constraint between constituent points. For a single pair of point masses, such a potential takes the form $V=\frac{1}{2}k(d(q_1,q_2)-\Delta)^2$ where $q_i$ are the locations of the masses, $d$ is the geodesic metric on $X$, and $\Delta$ is the rest length - the distance at which the potential is minimized. For a discrete set of points ${q_1,\ldots q_n}$ this naturally generalizes to $$V=\frac{1}{2}\sum_{ij}k_{ij}\left(d(q_i,q_j)-\Delta_{ij}\right)^2$$ for springs with spring constant $k_ij$ between the masses $q_i$ and $q_j$.
Taking the continuous limit provides a formula for a spring-like potential of general objects $$V=\frac{1}{2}\int_{(x,y)\in B\times B}(d(q(x),q(y))-\Delta(x,y))^2dk$$ where $dk$ is a spring density, which when specified as a sum of dirac measures reduces directly back to the finite sums given previously.

Dynamics: Because the internal physics has all been specified by a potential, the dynamics here are straightforward: we may directly utilize the unconstrained Euler-Lagrange equations on the space of embeddings. $$\frac{\partial \mathcal{L}}{\partial q^i}-\frac{\mathrm{d}}{\mathrm{dt}}\frac{\partial\mathcal{L}}{\partial v^i}=0$$