Physically Native World Models: A Hamiltonian Perspective on Generative World Modeling

World models enable model-based planning through learned latent dynamics, but imagined rollouts become unstable as the planning horizon grows or the dynamics distribution shifts. We argue that this instability reflects two missing structures in planner-facing latents: history-conditioned memory for approximate Markov completeness, and geometric organization that separates configuration, momentum, and task semantics. We propose HaM-World (HMW), a structured world model that decomposes the latent state into a canonical (q, p) subspace and a context subspace c, while using Mamba selective state-space memory as the history-conditioned input to the same latent dynamics. Within this interface, (q, p) evolves through an energy-derived Hamiltonian vector field plus learnable residual/control dynamics, while c captures semantic, dissipative, and non-conservative factors. This gives the planner a single latent state shared by dynamics prediction, reward/value estimation, imagined rollouts, and CEM action search. On four DeepMind Control Suite tasks, HaM-World reaches the highest Avg. AUC (117.9, +9.5%), reduces long-horizon rollout error to 45% of a strong baseline model, and wins 11/12 k in {3,5,7} MSE cells. Under 12 OOD perturbations spanning dynamics shifts, action delay, and observation masking, HaM-World achieves the highest return in every condition, with average OOD-return gains of 10.2% on Finger Spin and 13.6% on Reacher Easy. Mechanism diagnostics further show bounded action-free Hamiltonian-energy drift, structured energy variation under policy rollouts, and coherent control-induced energy transfer, supporting the intended Soft-Hamiltonian dynamics design.

Learning predictive world models from visual observations is a core problem in embodied AI, with applications to model-based reinforcement learning and robotic planning. Existing latent world models typically generate future states with unconstrained neural transition functions, while modern video generation systems often prioritize perceptual plausibility or introduce physical structure through auxiliary losses, external guidance, or separate dynamics modules. As a result, long-horizon rollouts can remain weakly grounded in the physical principles that govern real dynamics, leading to compounding error, energy drift, and physically inconsistent futures. We propose Least Action World Models (LaWM), a latent world-modeling framework that operationalizes the Principle of Least Action in learned visual latent space: future rollouts are governed by a learned Lagrangian action functional rather than produced only by an unconstrained transition predictor. Our main technical realization is a latent variational integrator: LaWM encodes observations into learned generalized coordinates, learns a latent discrete Lagrangian over consecutive latent states, constructs a discrete action functional, and advances prediction by solving the corresponding discrete integration condition. Thus, physical structure is not merely used to score, regularize, or constrain a completed trajectory; it defines the latent transition rule itself. Because the transition is induced by a discrete variational principle, LaWM provides a structure-preserving bias for long-horizon visual prediction. Across physics-clean synthetic dynamics and embodied robot interaction benchmarks, LaWM improves physical invariance, background consistency, motion smoothness, and appearance and geometric prediction metrics over video-generation and world-model baselines.

World models are increasingly used in embodied intelligence and generative simulation, yet their scope remains ambiguous across communities. This tutorial presents a design-space view of world models as action-conditioned predictive models that estimate the future evolution of task-relevant observations or states. We categorize existing methods into observation-space and state-space world models, comparing their trade-offs in visual fidelity, spatial structure, physical interpretability, and control usability. We further introduce world action models, which connect predicted futures with executable robot actions, and summarize four representative paradigms: imagine-then-execute, video-feature-conditioned action prediction, joint video-action modeling, and auxiliary video prediction for policy learning. The goal of this tutorial is to clarify the conceptual scope of world (action) models and provide a structured taxonomy for embodied prediction and control.