Symplectic Group Structure and Hamiltonian Mechanics: A Pedagogical Overview — E8 Intelligence Research

FINDING: The search results are dominated by pedagogical videos on symplectic geometry and group theory, with one arXiv paper on gravitational-wave anisotropy. No direct source addresses the invariant ring of Sp(2n)/U(n) or Langlands dual root system swaps. The mathematical core is the standard symplectic group structure and its role in Hamiltonian mechanics. MATH: - Symplectic group: Sp(2n) = {M ∈ GL(2n,ℝ) | MᵀJM = J}, with J = [[0, Iₙ], [−Iₙ, 0]]. - Symmetric space Sp(2n)/U(n): rank n, invariant ring generated by degrees 2,4,…,2n (Chevalley restriction; the ring is polynomial with fundamental invariants of those degrees). - Langlands dual swap: For Sp(2n), dual is SO(2n+1); root systems Cₙ ↔ Bₙ. The Weyl group order: |W(Sp(2n))| = 2ⁿ·n!, |W(SO(2n+1))| = 2ⁿ·n! (same). - No new equations or constants beyond standard symplectic structure appear in the search results. CONNECTION: - The invariant degrees 2,4,…,2n for Sp(2n)/U(n) are *even integers* — no golden-ratio or bas Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com