Physical Dynamics and Constraints › Physical Dynamics and Constraints

Imagine a robot arm that sweeps a cup off a table. A learned world model trained only on one-step prediction error can look convincing for a few frames and then produce an impossible rollout: the cup passes through the table, accelerates without a modeled force, or fails to slow when dissipation should matter. The loss did not explicitly require nonpenetration, force balance, or friction. These symptoms are useful diagnostic clues, but they are not one-to-one proofs of a missing concept. Penetration can also come from collision detection or integration error, for example. The practical question is which mechanical assumptions the model should enforce and which effects it should learn from data.

This chapter is about putting mechanics inside world models. We will work through rigid-body motion, deformable media, contact and friction, and conservation laws associated with symmetry. We will then study soft penalties, hard projections, and Hamiltonian architectures. A soft penalty discourages its encoded violation; a hard parameterization or verified projection can rule it out under stated solver assumptions. Exposed physical variables can also aid diagnosis when they are observable and identifiable. None of these choices guarantees accurate extrapolation: parameters, unmodeled forces, observations, and numerical integration can still be wrong.

As in Part I: The World-Model Idea, we keep a sharp line between what a model predicts and how an agent uses those predictions. Here we focus on the prediction side: making the dynamics model itself physically coherent. The decision-making machinery that consumes these predictions comes later, in Part VII: Planning and Agency. That separation is worth holding onto now, because it lets us evaluate a dynamics model on its own terms: as a predictor of what the world does, independent of any particular policy that might act on it.