Symplectic Geometry

*Verse 1* Symplectic geometry, a dance of forms, On differentiable realms, its grace transforms. Manifolds, curved and smooth, unfold their lore, With closed, nondegenerate 2-forms at their core. *Pre-Chorus* Hamiltonian whispers from classical days, Phase space waltzes, where symplectic lays. Systems in motion, their secrets unfurl, A symphony of structure, a manifold swirl. *Chorus* Symplectic, a calque, a linguistic embrace, Weyl's legacy woven, a complex grace. Latin's "com-plexus," braided threads unite, Greek's "sym-plektikos," entwined day and night. *Verse 2* Darboux's Theorem whispers secrets profound, Locally isomorphic, symplectic's ground. Standard vector spaces, their global refrain, Topological invariants, a celestial domain. *saxophone solo* *Bridge* Symplectic topology, a cosmic ballet, Global properties twirl, as geometry sways. Interchangeable terms, a harmonious blend, Symplectic's embrace, where dimensions transcend. *Outro* In this lyrical dance, where equations align, Symplectic's rhythm, a timeless design. From roots Indo-European, a tale to unfold, Complex and symplectic, their story retold. *Outro* In this lyrical dance, where equations align, Symplectic's rhythm, a timeless design. From roots Indo-European, a tale to unfold, Complex and symplectic, their story retold.