Geometric and Operatorial Perspectives in Hamiltonian and Relativistic Dynamics

The seminar outlines a unified operatorial approach to analytical dynamics in which geometric and algebraic structures are treated on equal footing. Beginning with the Jacobi-metric formulation of conservative systems, the Hamiltonian function is re-expressed through an alternative Legendre transformation that exposes hidden symmetries and separability patterns. Within this framework, the Levi-Civita separability condition is formulated as the vanishing of a five-argument operator constructed from nested Poisson brackets, providing a compact criterion for integrability.

The Deprit triangle of canonical perturbation theory is then interpreted in terms of Toeplitz-like operator matrices, clarifying the algebraic mechanisms underlying canonical reduction. Metric-based quantities such as the Einstein tensor and the Kretschmann scalar are presented as natural geometric invariants within the same operatorial language, bridging classical and relativistic dynamics. Finally, the framework is connected to applied astrodynamical problems through non-drift analytic conditions relevant to large-aperture satellite formations.

Together, these perspectives emphasize the emergence of an algebraic differential geometry unifying Hamiltonian theory, metric geometry, and operator analysis into a single coherent formalism