Computer Algebra Meets Hamiltonian Geometry

I describe a series of visually and algebraically compelling case studies that link computational algebra with the geometry of Hamiltonian and Lie–Poisson systems. Each example, fully reproducible in Maple, illustrates how symbolic computation can uncover the invariants, foliations, and algebraic varieties underlying Hamiltonian phase spaces. By computing Casimirs, symplectic leaves, and metriplectic extensions, one demonstrates how algebraic tools expose the topological and geometric structure of dynamical systems, insights that are increasingly essential for modern machine-learning approaches to physical dynamics and structure-preserving neural process models. This short compilation, intertwining algebra, and geometry is for my friend Professor Laureano González-Vega on the occasion of his 60th birthday, celebrating his academic research on real algebraic geometry, symbolic computation, and geometric modeling.