Symplectic group

In mathematics, the symplectic group is the group of linear transformations that preserve the geometric structure of phase space, the space of position and momentum variables used in classical mechanics. It is defined as the group of linear changes of coordinates on phase space that preserve the symplectic form.

The symplectic groups are usually denoted ${\displaystyle \operatorname {Sp} (2n,\mathbb {F} )}$, where ${\displaystyle n}$ is a positive integer and ${\displaystyle \mathbb {F} }$ is a field, often the real numbers or complex numbers. They are among the four families of classical groups and play a central role in symplectic geometry, Hamiltonian mechanics, and representation theory. A related but different family is the compact symplectic group, usually denoted ${\displaystyle \operatorname {Sp} (n)}$ or ${\displaystyle \mathrm {USp} (n)}$.