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Explore [Berkeley Seminar] Khyathi Komalan (Topos Institute) | Organising Compositionality in Mechanics with 2,290 views, 19 likes, and 2 comments. Experience the impact of this video content that has captured audience attention.

Title: Organising Compositionality in Classical Mechanics Abstract: Consider a scenario where we give two people each two springs and ask them to build something with them. There is no reason to believe that the resulting systems would be the same — one may choose to make the springs share a common mass, while another may decide to leave them separate. While the components remain the same, the resulting system does not. Mathematically, where exactly should the choice of how to put things together live? Baez, Weisbart and Yassine think about this by modelling open classical systems as augmented spans and composing compatible systems using pullbacks. Their construction keeps track of Lagrangian and Hamiltonian descriptions of mechanics — the former using positions and velocities, the latter positions and momenta — but doing so requires a fair amount of machinery, including separate Riemannian and Poisson settings and a generalized F-pullback construction. I’ll reorganise this picture by working with smooth spaces carrying closed 2-forms, where ordinary pullbacks are available. These 2-forms encode the geometry of position–momentum phase space, and our systems are required to respect that geometry. For the spring examples, differentiating the potential picks out a half-dimensional subspace on which the 2-form vanishes — called a Lagrangian submanifold. The resulting span theory then keeps the systems themselves separate from the ways of assembling them. So we can fix two springs once and vary only the instruction for how they should be combined. Most of the talk will involve seeing this in action through examples involving springs. Tensoring two spring systems simply places them side by side, without yet deciding how they are connected. That extra choice lives in a loose span, whose action by pullback performs the assembly. Changing the span while keeping the springs fixed can therefore produce different mechanical systems. We’ll work this out explicitly for two springs sharing a mass, and end by asking whether the same mechanism can also compose Hamiltonian evolution through time. Date: 8/25/26

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