It is a rarely appreciated fact that in $3$ dimensions, every rigid solid has a state of equilibrium. Here's what I mean: if I roll a dice on a flat table, I can be sure that after a certain amount of time it will achieve a state of relative stability and stop moving. This notion of “relative stability” is what I call the state of equilibrium.
To anyone familiar with physics it is obvious that this notion is equivalent in some sense to some local minimum, namely a local minimum in gravitational potential energy. In fact, to the physicist this property is a priori true by virtue of the principle of least potential energy, which states that any physical system will converge over time to a state where total potential energy is minimised.
However the principle of least potential energy is just that: a principle. It simply assumes the existence of such states. In this post I will attempt to give a purely mathematical perspective by rigorously defining each notion in order to see if the property holds mathematically in $n$ dimensions, and perhaps even in other vector spaces.
Let us begin by defining solid. This definition should encapsulate both how a solid can be shaped, and how its mass can be distributed in it.
Definition 1: A solid $\mathcal{S}$ in $n$ dimensions is a compact, connected set $S \subseteq \mathbb{R}^n$ equipped with a density function $\rho:S\to\mathbb{R}_{>0}$. We say that $\mathcal{S}=(S,\rho).$
(For the sake of realism it would make sense to impose some constraints on $\rho$, because the mass distribution of a real-life object cannot be totally arbitrary. Let us say that $\rho$ must be piecewise continuous in the sense that if any path $\Gamma \subset S$ is parametrised by $\gamma :[a,b] \to \Gamma$, then $\rho(\gamma(t))$ is piecewise continuous on $[a,b]$.)
From this first definition we can intuitively derive the notion of mass:
Definition 2: The mass of $\mathcal{S}$ is defined by $$M(\mathcal{S})=\int_S \rho(s)\ ds.$$
Next it would be of interest to define gravitational potential energy since this is equilibrium points are most concerned with. Before doing so, let us agree to say that given some point $x \in \mathbb{R}^n$, the height $h(x)$ of $x$ is the value of its $n$-th coordinate. For example, the height of the point $(1,-1,2)\in \mathbb{R}^3$ is $2$; the height of $(1,0)\in \mathbb{R}^2$ is $0$, etc.
Definition 3: The gravitational potential energy of $\mathcal{S}$ is defined by $$E(\mathcal{S})=\int_S h(s)\rho(s)\ ds.$$
For the previous definition to have a physical meaning the points of $\mathcal{S}$ must all have a positive height. We will say that $\mathcal{S}$ is positive iff for all $s \in S$, $h(s) \geqslant 0$.
We can now move on to defining a notion of equivalence between solids. More precisely, we want to introduce the idea that a solid is still essentially the same if it is shifted or rotated in any direction.
Definition 4: Let $\mathcal{S}=(S,\rho)$ and $\mathcal{T}=(T,\tau)$ be solids. A function $F:S \to T$ is called a translation iff it is:
(a) bijective
(b) distance-preserving, i.e. for any $s,s' \in S$, $\|F(s)-F(s')\|=\|s-s'\|$
(c) density-preserving, i.e. for any $s \in S$, $\tau(F(s))=\rho(s)$.
Definition 5: Let $\mathcal{S}=(S,\rho)$ and $\mathcal{T}=(T,\tau)$ be solids. If there exists a function $F:S \to T$ that is a translation, we say that $\mathcal{S}$ and $\mathcal{T}$ are similar and we write $\mathcal{S} \sim \mathcal{T}$.
From these definitions we derive the following results:
Proposition 1: If $\mathcal{S} \sim \mathcal{T}$ then $M(\mathcal{S})=M(\mathcal{T})$.
Proposition 2: The binary relation $\sim$ is an equivalence relation over the set of solids in $\mathbb{R}^n$.
Now the scene is set for us to be able to define the $\varepsilon$-neighborhood of a solid, which is crucial in defining the local minimum of potential energy:
Definition 6: Let $\mathcal{S}=(S,\rho)$ be a solid and $\varepsilon > 0$. The $\varepsilon$-neighborhood $V_\varepsilon(\mathcal{S})$ of $\mathcal{S}$ is defined as the set of solids $\mathcal{W}$ which are similar to $\mathcal{S}$ under some translation $F$ such that $$\int_S \|F(s) - s\|\ ds < \varepsilon.$$
Finally, since we prefer working with positive-definite solids we will set one final definition before moving on to the big question.
Definition 7: Given a solid $\mathcal{S}$, the positive $\varepsilon$-neighborhood $V_\varepsilon^+(\mathcal{S})$ is the set of solids in $V_\epsilon(\mathcal{S})$ that are positive.
In this context the question of the existence of equilibrium states can be posed as follows:
Proposition 3: Let $\mathcal{S}$ be a solid. Then there exists a positive solid $\mathcal{S'}$ with $\mathcal{S'} \sim \mathcal{S}$ such that for some $\varepsilon > 0$, $$E(\mathcal{S}') \leqslant E(\mathcal{T}) \quad \textrm{for every } \mathcal{T} \in V_\varepsilon^+(\mathcal{S}').$$
In order of strength, my questions are as follows:
Is Proposition 3 true in $\mathbb{R}^2$ and $\mathbb{R}^3$?
Is Proposition 3 true in $\mathbb{R}^n$?
Can Proposition 3 be generalised to other vector spaces?