symplectic form

A symplectic form is an alternating bilinear form that is also nondegenerate: it satisfies B(v, v) = 0 for all v (alternating) and has trivial radical (nondegenerate). It is the geometry of signed area rather than length — the natural structure for phase space in classical mechanics, where pairs (position, momentum) live and area in each such pair is conserved by the dynamics.

Two facts pin symplectic forms down tightly. First, they only exist in even dimension: an alternating form has even rank, and nondegeneracy forces the rank to equal the full dimension, so n must be even. Second, there is no signature to classify them — over any field, every symplectic form looks the same in a suitable basis. All symplectic spaces of a given dimension are isomorphic.

That canonical model is the standard symplectic matrix J, built from hyperbolic blocks [0, 1; -1, 0] stacked down the diagonal (equivalently the block matrix [0, I; -I, 0]). A Darboux-style basis splits coordinates into conjugate pairs (q_i, p_i) with B(q_i, p_j) = delta_ij and all other pairings zero — the algebraic skeleton of Hamilton's equations.

The linear maps preserving a symplectic form are the symplectic group Sp(2n), the alternating-form cousin of the orthogonal group that preserves a symmetric inner product. Where orthogonal maps preserve length and the spectral theorem, symplectic maps preserve area and underlie Hamiltonian flows, Liouville's theorem, and a great deal of modern geometry. This term is the preview of that whole world.