symplectic group Sp(2n)

Orthogonal matrices preserve a SYMMETRIC form (the inner product, which feeds you lengths). Symplectic matrices preserve an ALTERNATING form instead — a pairing that measures oriented AREA rather than length. The symplectic group is the natural symmetry group of classical mechanics, where position and momentum pair up by exactly such a form.

Fix the standard symplectic form omega(x, y) = x^T J y, where J = [0, I; -I, 0] is the 2n x 2n block matrix. Then Sp(2n, F) = { A : A^T J A = J }. This is the alternating-form analog of the orthogonal condition A^T A = I; you have just swapped the symmetric I for the skew-symmetric J. The matrices live in even dimension 2n because an alternating form needs paired coordinates.

Geometrically, a symplectic map preserves omega, hence preserves all the oriented 2-dimensional areas omega measures. In Hamiltonian mechanics, time evolution of a system is symplectic — Liouville's theorem (phase-space volume is conserved) is one consequence, because preserving omega forces det A = 1, so Sp(2n) sits inside SL(2n).

A caveat on what is and is not preserved: symplectic maps do NOT in general preserve lengths or angles — Sp(2n,R) is non-compact and much larger than O(2n). The only place the three worlds meet is the 'unitary trick': Sp(2n,R) intersect O(2n) is isomorphic to U(n), a fact that links symplectic geometry back to complex structure.