“Everything is a Lagrangian submanifold...”

Reality as geometry

Alan Weinstein said, “Everything is a Lagrangian submanifold...”

In other words, forget particles. Forget waves.

To him, the universe is built from something entirely different: Lagrangian submanifolds. What are those, and why should you care? To grasp Lagrangian submanifolds, you first need to know about phase space…

Usually, people say phase space (incorrectly) when they actually mean “state space,” so let’s define this phase space. It’s an abstract space (not spacetime) where each point represents a particle’s state given by its position q and momentum p. So, phase space is a space of (q, p) pairs. You can extend this to N particles, of course.

For now, we just have a space without geometry. You could put a certain type of metric, making it Riemannian, or you can place something else and make it “symplectic.” We take this latter route. This equips phase space with a special structure, a 2-form, denoted by ω. This ω lets you define areas in phase space and, importantly, dictates how systems evolve in time. The rule is elegantly simple:

\(\{f, H\} = \frac{df}{dt}\)

Here, H is the Hamiltonian (the energy of the system), and {f, H} is the Poisson bracket, which uses ω to measure how a function f changes along the flow generated by H. So, if {f, H} = 0, then f is a conserved quantity. Time then must satisfy {t, H} = 1. This is a beautiful condition of time (time must evolve linearly with respect to itself).

Okay, so what’s a Lagrangian submanifold then? It’s a special kind of subspace L within phase space where the symplectic form ω vanishes. In elementary terms, it’s a subspace where “areas,” as measured by ω, are always zero. They often represent physically meaningful sets of states.

Weinstein’s quip is that these Lagrangian submanifolds are the building blocks of nature more so than particles or waves. In classical mechanics, the system’s actual trajectory through phase space is itself a 1-dimensional Lagrangian submanifold (if you have a 2-dimensional symplectic manifold).

This means the system evolves in a way that preserves these “symplectic areas.” This is why geometry is tied to physics. The ω was originally thought to be about areas, but now you can see it’s about dynamics.

Now, most of the time in physics (in life as well), you have constraints. If a system is constrained (like a pendulum restricted to a certain length), its possible states may lie on a Lagrangian submanifold. In fact, many physical constraints can be expressed in this geometric language.

Even quantization, which is often ill-defined but at least stated aloud as the jump from classical to quantum theory, can also be viewed through Lagrangian submanifolds. Certain “quantization conditions” pick out specific Lagrangian submanifolds that correspond to allowed quantum states.

More will be explored shortly here on a podcast with Eva Miranda.

So, Weinstein’s idea that “everything is a Lagrangian submanifold” actually has something to it. It’s a claim that the fundamental objects of reality aren’t particles or waves but these special subspaces within phase space that encode the system’s dynamics, constraints, and even its quantum nature.

Some see this as a shift from “things” to relationships (especially categorists, or Matt Segall) -- the relationships between position and momentum, energy and time, as encoded in the symplectic geometry. And these relationships are (often) captured by Lagrangian submanifolds.

- Curt Jaimungal