Physical Dynamics and Constraints › Summary
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Physical dynamics and constraints provide strong, testable priors for embodied world models. The key takeaways:
- Rigid-body dynamics reduces configuration to generalized coordinates. For regular conservative systems, Lagrangian or Hamiltonian formulations derive the dynamics from a scalar function; identifying the Hamiltonian with physical total energy requires the usual autonomous natural-system assumptions.
- Deformable media require many degrees of freedom. Continuum-based schemes such as FEM and MPM and discrete schemes such as mass–spring systems and PBD make different commitments about material behavior.
- Contact and friction introduce unilateral constraints, impulses, and stick–slip mode changes. A smooth predictor may approximate them, but exact nonpenetration or friction feasibility requires explicit enforcement or a verified solver.
- Noether's theorem links qualifying continuous variational symmetries to conserved quantities. Symplectic updates preserve a two-form, while equivariant architectures enforce one selected group action; neither automatically preserves every physical invariant.
- Enforcement is a spectrum: soft penalties discourage violations, structured architectures enforce specified properties, projections impose selected feasibility conditions to solver tolerance, and hybrid residual models combine a backbone with learnable mismatch. Each guarantee is conditional on the encoded law, state, and numerical method.
The pendulum experiment gives a narrower lesson: structural conservation and predictive accuracy must be checked separately. The HNN nearly conserves its learned scalar and has the lower 20-second endpoint error on this held-out rollout after both models are trained on the same analytic vector field, but a single seed cannot establish a general architecture ranking. Controlled mechanical models can represent how actions enter the dynamics; passive observational data may not identify those intervention effects without additional assumptions. The next chapter develops causal models for making that distinction explicit.