Symplectic vector spaces

AUTHORS:

  • Tobias Diez (2021): initial version

class sage.manifolds.differentiable.examples.symplectic_space.StandardSymplecticSpace(dimension: int, name: str | None = None, latex_name: str | None = None, coordinates: str = 'Cartesian', symbols: str | None = None, symplectic_name: str | None = 'omega', symplectic_latex_name: str | None = None, start_index: int = 1, base_manifold: StandardSymplecticSpace | None = None, names: tuple[str] | None = None)[source]

Bases: EuclideanSpace

The vector space \(\RR^{2n}\) equipped with its standard symplectic form.

symplectic_form()[source]

Return the symplectic form.

EXAMPLES:

Standard symplectic form on \(\RR^2\):

sage: M.<q, p> = manifolds.StandardSymplecticSpace(2, symplectic_name='omega')
sage: omega = M.symplectic_form()
sage: omega.display()
omega = -dq∧dp

>>> from sage.all import *
>>> M = manifolds.StandardSymplecticSpace(Integer(2), symplectic_name='omega', names=('q', 'p',)); (q, p,) = M._first_ngens(2)
>>> omega = M.symplectic_form()
>>> omega.display()
omega = -dq∧dp