I wrote a proof from first principles that energy is conserved in a $D$-dimensional rotating rigid body without external forces, and I'd like to ask for some feedback on improving my math with more idiomatic notation, perhaps making use of geometric algebra; my background is in machine learning, meaning I tend to do a lot of linear algebra and statistics.
Say we have a $D$-dimensional rigid body composed of $N$ points of equal mass with coordinates $\mathbf x_i \ \forall i\in\{1\dots N\}$, vectors which together form the matrix $\mathbf X_{D \times N}$. These points are moving and have some velocities and accelerations, similarly forming the matrices $\mathbf V_{D \times N}$ and $\mathbf A_{D \times N}$ respectively.
For the sake of simplicity (and without loss of generality for rotational kinetic energy conservation), let's assume the center of mass of the body is fixed at the origin, and add the assumption that we have no outside forces. Then we can write: $$\mathbf{X1}=\mathbf 0 \\ \mathbf{V1}=\mathbf 0\\ \mathbf{A1}=\mathbf 0$$ where in these equations, $\mathbf 1$ and $\mathbf 0$ are vectors composed of all ones and all zeros respectively.
There are three "first-principles" I will use:
- constant distances between point pairs: $ \frac{\mathrm d}{\mathrm d t} \| \mathbf x_i - \mathbf x_j \| =0 \ \forall i,j $
- all forces are central: $ \mathbf a_i = \sum_j k_{ij} (\mathbf x_i - \mathbf x_j) $
- Newton's (strong?) 3rd Law: $ k_{ij} = k_{ji} $
I'll go over the main results without going into too much detail with the algebra. First, I define the matrices:
$$\mathbf R_{N \times N} = \mathbf{X^T X} \\ \mathbf J_{D \times D} = \mathbf{X X^T} \\ \mathbf L_{D \times D} = \mathbf{X V^T - V X^T} $$
The matrix $\mathbf J$ is homologous to the moment of inertia, but since I'm not using cross-products, I end up with a simpler form. And the matrix $\mathbf L$ is homologous to the angular momentum, which I've recently learned I might be able to write using a wedge product as $$\mathbf L \propto \left<\mathbf x \wedge \mathbf v \right> $$ where I use the stats notation $\left<\mathbf x \right> = \frac{1}{N} \sum_{i=1}^n{\mathbf x_i}$. And I can prove angular momentum is conserved from Newton's 3rd Law and the force centrality.
Here is the first problem I'm encountering: while this pattern with the minus looks like a wedge product in geometric algebra, I also get a lot of a similar pattern but with plus instead of minus; in particular, I use the fact that $$\frac{\mathrm d}{\mathrm d t}\mathbf R = \mathbf{X^T V} + \mathbf{V^T X} = \mathbf 0$$ to prove that there exists a unique skew-symmetric matrix $\mathbf W_{D \times D}$ such that $\mathbf V = \mathbf {WX}$, which I use to represent angular velocity. Plus, I make use of both $[\mathbf{X^T X}]_{ij} = \mathbf{x}_i \cdot \mathbf{x}_j $ as well as $ \mathbf{X X^T} \propto \left<\mathbf{x x^T} \right> $ in my equations, but from what I've read so far, geometric algebra doesn't have both an inner product and an outer product in addition to the wedge product, so I'm unsure how to make all my equation work in the framework of geometric algebra.
Continuing the outline of my proof, I show that $$\mathbf L = -(\mathbf{JW}+\mathbf{WJ}) = \mathbf{JW^T}-\mathbf{WJ^T} $$ I guess this one I can write both as a sum or as a difference, since $\mathbf J$ is symmetric, and $\mathbf W$ is skew-symmetric.
Since angular momentum is conserved, $$\dot{\mathbf L} = -\mathbf{\dot{J}W}-\mathbf{J\dot{W}} - \mathbf{\dot{W}J}-\mathbf{W\dot{J}} = \mathbf 0$$ and I can use this to solve for the angular acceleration $\dot{\mathbf W}$, as I get the Sylvester equation $$\mathbf{J\dot{W}} + \mathbf{\dot{W}J} = -\mathbf{\dot{J}W}-\mathbf{W\dot{J}}$$
Alternatively, I could use the pattern with minuses to write
$$\mathbf{J\dot{W}^T} - \mathbf{\dot{W}J^T} = \mathbf{\dot{J}W^T}-\mathbf{W\dot{J}^T}$$
but this is not a Sylvester equation anymore, and I don't know how to solve it. Is there an idiomatic way to write this equation using wedge products? And how would I solve it once it's in terms of wedge products?
Finally, I can complete my proof of energy conservation by showing that I can compute the total energy of the system using the matrix trace in the equation $$\mathrm{Tr}[\mathbf{LW}]=2\mathrm{Tr}[\mathbf{V^T V}]$$ then taking the derivative of the left side to show that $$\frac{\mathrm d}{\mathrm d t}\mathrm{Tr}[\mathbf{LW}]=\mathrm{Tr}[-\mathbf{(WW^T)L}]=0$$ since the trace of the product of a symmetric matrix with a skew-symmetric matrix is zero, but I'm not using the wedge-like looking pattern of something minus its transpose anymore. And I can find the acceleration matrix for the individual particles with $$\mathbf{A} = \mathbf{WV} + \dot{\mathbf{W}}\mathbf{X}$$
My two questions are:
- Can I rewrite all of this using geometric algebra, despite making use of both inner and outer products?
- What happens to my Sylvester equation once I use geometric algebra? How do I solve it?