⇤Math diary
Aug 29, 2023: Euler's euqation is wacky
Consider a fluid on $\mathbb{R}^n$ (with coordinate $\vec{x}$), evolving with time (with coordinate $t$). How does the velocity of the fluid change over time? This falls under the purview of fluid mechanics, where the simplest evolution is described by the incompressible euler’s equations: \(\begin{align} \frac{Du}{Dt} &= \nabla P \\ \nabla \cdot u &= 0 \end{align}\)Here $u$ is the velocity field of the fluid, and $P$ is the pressure. $\frac{Du}{Dt}$ represents the advective derivative of velocity. It represents how much the velocity changes, as you’re moving with the velocity. Said another way, let $\phi_t$ be the flow of the dependent velocity field $u(t)$. For a quantity $f(x,t)$, suppose you measured this quantity as it flowed with the fluid. The quantity would change with time as $\phi_t^* f(x,t)$. The time derivative of this has two contributions, from the change in $f$ with space induced by the velocity of the fluid, and from the change in $f$ with time. In particular, \(\frac{Df}{Dt} = \partial_t \phi_t^* f(x,t) = \partial_t f + \cal{L}_u f = \partial_t f + \vec{u} \cdot \nabla f\)The derivative of the pullback is by definition the lie derivative, which can be written in coordinates as the second term in the rightmost expression.
Euler’s equations are so simple and fundamental, that they have many seemingly unrelated derivations.
Derivation 1: physics
This one’s pretty straightforward. Imagine each parcel of fluid a position $x$ is its own little guy, moving with velocity $u(x)$. Newtons laws say that the acceleration of the fluid equals the force applied. Force is just the gradient of pressure, and acceleration is $\frac{Du}{Dt}$, so this becomes the first of euler’s equations. Next we impose that the fluid is incompressible, which is captured by saying the velocity is divergence-free. And thats euler’s equation!
Derivation 2: geodesics
My favorite derivation hails from V. Arnold. As the fluid flows over time, its configuration is encoded as a diffeomorphism of $\mathbb{R}^n$. For an incomporessible fluid, this is a volume-preserving diffeomorphism. The set of all volume preserving diffeomorphisms forms an infinite-dimensional lie group. The geodesics on this lie group (according to its unique invariant metric) are the euler equations!
Derivation 3: Hamiltonians
This one’s weird, and why I wanted to write this entry. In my symplectic geometry class today, the professor mentioned this as an application of Poincare’s invariant. The recipe is, consider a hamiltonian on phase space $\mathbb{R}^{2n}$, with coordinates $q,p$. Suppose the Hamiltonain is $H(q,p) = \vert p \vert ^2 + P(q)$ for the pressure $P$. Suppose you can find a velocity field $u(q)$ such that the flow lines of this velocity field are exactly hamiltonian trajectories for the supplied hamiltonian. This means we have two sets of equations: Hamilton’s equations, and the condition $p = u(q)$. For these diffeqs to have a solution, $u$ must satisfy a consistency condition. This is exactly euler’s equation. In other words, Eulers equation exactly persecribe the vector fields whose tranjectories are hamiltonian trajectories of particles under the pressure potential.
But why should euler’s equations give hamiltonian-like evolution? This is an accident of specifically incompressible Euler’s equations. The argument I heard is, we can take a microscopic limit, where the configuration space of the fluid has dimension of order $10^{23}$. This is hamiltonian evolution. Tracing any individual water moluacle, we see it moves in a very chaotic path. But statistically, averaging over close particles, the moluacle roughly travels in flowlines of the mascroscopic velocity field. It so happens that, in the statistical average, it still acts as a hamiltonian flow. Thus, 10^23 dimensions reduces to 6!