Observables in Motion: A guide to simulating classical and quantum dynamics

We present a pedagogical work-in-progress. This textbook aims to introduce Hilbert space representations for quantum and classical dynamics, outlining the mathematical foundations, practical guidance, and Python implementation of dynamical simulations. Beginning with a historical survey, the book systematically develops the mathematical foundations, kinematic descriptions, and formal evolution theory needed to understand both classical and quantum systems. It then provides practical guidance for numerically simulating time-dependent Schrödinger evolution, classical dynamics through symplectic integrators, and open quantum systems including phase-space formulations and stochastic unravellings.