# Classical dynamics of a matrix

For a system of interacting particles, we can formulate Hamiltonian dynamics in terms of a vector of position coordinates $q$ and a vector of momentum coordinates $p$. Then the Hamiltonian takes the form

$$H(p,q) = \frac{1}{2m}p^2+V(q)$$

One can show that the dynamics following from Hamilton's equations obey certain symmetries. For example, the dynamics are symplectic (phase-space volume preserving). Also, for a $V(q)$ that has no explicit time dependence, the energy will be conserved and there will be time-reversal symmetry.

My question is, what if we consider the Hamiltonian or Lagrangian dynamics of a matrix? Perhaps we might use a Hamiltonian of the form

$$H(P,Q)=\frac{1}{2m}||P||_F^2+V(Q)$$

where $P$ and $Q$ are matrices and I have used the Frobenius norm. Are there any new symmetries that emerge here that we couldn't consider for vector dynamics? Is there any sort of conservation law regarding determinants or traces of matrices involving $P$ and $Q$? How about the relation among the columns of $P$ and $Q$? If the columns start out linearly independent, will they stay linearly independent?

Here's a specific example of a potential:

$$V(Q)=\frac{m\omega^2}{2}||Q-A(Q)||_F^2$$

where $A$ is a possibly $Q$-dependent matrix. When $A$ is constant this is something like a harmonic oscillator potential.

Has anyone written any papers or books about classical dynamics of a matrix? If someone could point me in the right direction I'd be grateful.

• Aren't you just forming special expressions of your generalized coordinates? – CuriousOne May 27 '16 at 5:57
• That's one way to think about it but are there any symmetries like I mention above? – Ian May 27 '16 at 13:42
• I am not saying that you are not up to something, matrices are characterized by a number of invariants, like their trace and determinants, so it might be that there is an interesting relationship there, I just can't see it, but that's because I haven't done theory in ages, it's not a problem with your question. – CuriousOne May 27 '16 at 19:02

There is a well-known isomorphism between the linear space ${\mathcal M}_{m, n}$ of $m\times n$ matrices and typical (vectorial) linear spaces ${\mathcal L}_{m\times n}$ of dimension $\text{dim}({\mathcal L}_{m\times n}) = m\times n$. Everything that is valid in ${\mathcal L}_{m\times n}$ has an equivalent in ${\mathcal M}_{m, n}$ and conversely. For this reason there may not be much of a qualitative difference between a matrix dynamics of the type you propose and the standard versions formulated in terms of vector variables.
Personally I do not recall any major applications of this isomorphism in classical mechanics, but nowadays it is heavily used in quantum theory in relation to density matrices and the associated superoperator formalism: quantum information theory, quantum entanglement and separability, open quantum systems, etc. The isomorphism is established by casting any $m\times n$ matrix $M$ as a $m\times n$ vector $\textbf{vec}(M)$ defined by concatenating $M$'s columns. See for instance this paper (arXiv).