Symplectic structures#

The class SymplecticForm implements symplectic structures on differentiable manifolds over \(\RR\). The derived class SymplecticFormParal is devoted to symplectic forms on a parallelizable manifold.

AUTHORS:

Tobias Diez (2021) : initial version

REFERENCES:

class sage.manifolds.differentiable.symplectic_form.SymplecticForm(manifold: Union[DifferentiableManifold, VectorFieldModule], name: Optional[str] = None, latex_name: Optional[str] = None)#

Bases: DiffForm

A symplectic form on a differentiable manifold.

An instance of this class is a closed nondegenerate differential \(2\)-form \(\omega\) on a differentiable manifold \(M\) over \(\RR\).

In particular, at each point \(m \in M\), \(\omega_m\) is a bilinear map of the type:

\[\omega_m:\ T_m M \times T_m M \to \RR,\]

where \(T_m M\) stands for the tangent space to the manifold \(M\) at the point \(m\), such that \(\omega_m\) is skew-symmetric: \(\forall u,v \in T_m M, \ \omega_m(v,u) = - \omega_m(u,v)\) and nondegenerate: \((\forall v \in T_m M,\ \ \omega_m(u,v) = 0) \Longrightarrow u=0\).

Note

If \(M\) is parallelizable, the class SymplecticFormParal should be used instead.

INPUT:

manifold – module \(\mathfrak{X}(M)\) of vector fields on the manifold \(M\), or the manifold \(M\) itself
name – (default: omega) name given to the symplectic form
latex_name – (default: None) LaTeX symbol to denote the symplectic form; if None, it is formed from name

EXAMPLES:

A symplectic form on the 2-sphere:

sage: M.<x,y> = manifolds.Sphere(2, coordinates='stereographic')
sage: stereoN = M.stereographic_coordinates(pole='north')
sage: stereoS = M.stereographic_coordinates(pole='south')
sage: omega = M.symplectic_form(name='omega', latex_name=r'\omega')
sage: omega
Symplectic form omega on the 2-sphere S^2 of radius 1 smoothly embedded
 in the Euclidean space E^3

omega is initialized by providing its single nonvanishing component w.r.t. the vector frame associated to stereoN, which is the default frame on M:

sage: omega[1, 2] = 1/(1 + x^2 + y^2)^2

The components w.r.t. the vector frame associated to stereoS are obtained thanks to the method add_comp_by_continuation():

sage: omega.add_comp_by_continuation(stereoS.frame(),
....:                  stereoS.domain().intersection(stereoN.domain()))
sage: omega.display()
omega = (x^2 + y^2 + 1)^(-2) dx∧dy
sage: omega.display(stereoS)
omega = -1/(xp^4 + yp^4 + 2*(xp^2 + 1)*yp^2 + 2*xp^2 + 1) dxp∧dyp

omega is an exact 2-form (this is trivial here, since M is 2-dimensional):

sage: diff(omega).display()
domega = 0

flat(vector_field)#

Return the image of the given differential form under the map \(\omega^\flat: T M \to T^*M\) defined by

\[<\omega^\flat(X), Y> = \omega_m (X, Y)\]

for all \(X, Y \in T_m M\).

In indices, \(X_i = \omega_{ji} X^j\).

INPUT:

vector_field – the vector field to calculate its flat of

EXAMPLES:

sage: M = manifolds.StandardSymplecticSpace(2)
sage: omega = M.symplectic_form()
sage: X = M.vector_field_module().an_element()
sage: X.set_name('X')
sage: X.display()
X = 2 e_q + 2 e_p
sage: omega.flat(X).display()
X_flat = 2 dq - 2 dp

hamiltonian_vector_field(function)#

The Hamiltonian vector field \(X_f\) generated by a function \(f: M \to \RR\).

The Hamiltonian vector field is defined by

\[X_f \lrcorner \omega + df = 0.\]

INPUT:

function – the function generating the Hamiltonian vector field

EXAMPLES:

sage: M = manifolds.StandardSymplecticSpace(2)
sage: omega = M.symplectic_form()
sage: f = M.scalar_field({ chart: function('f')(*chart[:]) for chart in M.atlas() }, name='f')
sage: f.display()
f: R2 → ℝ
   (q, p) ↦ f(q, p)
sage: Xf = omega.hamiltonian_vector_field(f)
sage: Xf.display()
Xf = d(f)/dp e_q - d(f)/dq e_p

hodge_star(pform)#

Compute the Hodge dual of a differential form with respect to the symplectic form.

See hodge_dual() for the definition and more details.

INPUT:

pform: a \(p\)-form \(A\); must be an instance of DiffScalarField for \(p=0\) and of DiffForm or DiffFormParal for \(p\geq 1\).

OUTPUT:

the \((n-p)\)-form \(*A\)

EXAMPLES:

Hodge dual of any form on the symplectic vector space \(R^2\):

sage: M = manifolds.StandardSymplecticSpace(2)
sage: omega = M.symplectic_form()
sage: a = M.one_form(1, 0, name='a')
sage: omega.hodge_star(a).display()
*a = dq
sage: b = M.one_form(0, 1, name='b')
sage: omega.hodge_star(b).display()
*b = dp
sage: f = M.scalar_field(1, name='f')
sage: omega.hodge_star(f).display()
*f = -dq∧dp
sage: omega.hodge_star(omega).display()
*omega: R2 → ℝ
   (q, p) ↦ 1

on_forms(first, second)#

Return the contraction of the two forms with respect to the symplectic form.

The symplectic form \(\omega\) gives rise to a bilinear form, also denoted by \(\omega\) on the space of \(1\)-forms by

\[\omega(\alpha, \beta) = \omega(\alpha^\sharp, \beta^\sharp),\]

where \(\alpha^\sharp\) is the dual of \(\alpha\) with respect to \(\omega\), see up(). This bilinear form induces a bilinear form on the space of all forms determined by its value on decomposable elements as:

\[\omega(\alpha_1 \wedge \ldots \wedge\alpha_p, \beta_1 \wedge \ldots \wedge\beta_p) = det(\omega(\alpha_i, \beta_j)).\]

INPUT:

first – a \(p\)-form \(\alpha\)
second – a \(p\)-form \(\beta\)

OUTPUT:

the scalar field \(\omega(\alpha, \beta)\)

EXAMPLES:

sage: M = manifolds.StandardSymplecticSpace(2) sage: omega = M.symplectic_form() sage: a = M.one_form(1, 0, name=’a’) sage: b = M.one_form(0, 1, name=’b’) sage: omega.on_forms(a, b).display() R2 → ℝ (q, p) ↦ -1

poisson(expansion_symbol=None, order=1)#

Return the Poisson tensor associated with the symplectic form.

INPUT:

expansion_symbol – (default: None) symbolic variable; if specified, the inverse will be expanded in power series with respect to this variable (around its zero value)
order – integer (default: 1); the order of the expansion if expansion_symbol is not None; the order is defined as the degree of the polynomial representing the truncated power series in expansion_symbol; currently only first order inverse is supported

If expansion_symbol is set, then the zeroth order symplectic form must be invertible. Moreover, subsequent calls to this method will return a cached value, even when called with the default value (to enable computation of derived quantities). To reset, use _del_derived().

OUTPUT:

the Poisson tensor, as an instance of PoissonTensorField()

EXAMPLES:

Poisson tensor of \(2\)-dimensional symplectic vector space:

sage: M = manifolds.StandardSymplecticSpace(2)
sage: omega = M.symplectic_form()
sage: poisson = omega.poisson(); poisson
2-vector field poisson_omega on the Standard symplectic space R2
sage: poisson.display()
poisson_omega = -e_q∧e_p

poisson_bracket(f, g)#

Return the Poisson bracket

\[\{f, g\} = \omega(X_f, X_g)\]

of the given functions.

INPUT:

f – function inserted in the first slot
g – function inserted in the second slot

EXAMPLES:

sage: M.<q, p> = EuclideanSpace(2)
sage: poisson = M.poisson_tensor('varpi')
sage: poisson.set_comp()[1,2] = -1
sage: f = M.scalar_field({ chart: function('f')(*chart[:]) for chart in M.atlas() }, name='f')
sage: g = M.scalar_field({ chart: function('g')(*chart[:]) for chart in M.atlas() }, name='g')
sage: poisson.poisson_bracket(f, g).display()
poisson(f, g): E^2 → ℝ
   (q, p) ↦ d(f)/dp*d(g)/dq - d(f)/dq*d(g)/dp

restrict(subdomain, dest_map=None)#

Return the restriction of the symplectic form to some subdomain.

If the restriction has not been defined yet, it is constructed here.

INPUT:

subdomain – open subset \(U\) of the symplectic form’s domain
dest_map – (default: None) smooth destination map \(\Phi:\ U \to V\), where \(V\) is a subdomain of the symplectic form’s domain If None, the restriction of the initial vector field module is used.

OUTPUT:

the restricted symplectic form.

EXAMPLES:

sage: M = Manifold(6, 'M')
sage: omega = M.symplectic_form()
sage: U = M.open_subset('U')
sage: omega.restrict(U)
2-form omega on the Open subset U of the 6-dimensional differentiable manifold M

sharp(form)#

Return the image of the given differential form under the map \(\omega^\sharp: T^* M \to TM\) defined by

\[\omega (\omega^\sharp(\alpha), X) = \alpha(X)\]

for all \(X \in T_m M\) and \(\alpha \in T^*_m M\). The sharp map is inverse to the flat map.

In indices, \(\alpha^i = \varpi^{ij} \alpha_j\), where \(\varpi\) is the Poisson tensor associated with the symplectic form.

INPUT:

form – the differential form to calculate its sharp of

EXAMPLES:

sage: M = manifolds.StandardSymplecticSpace(2)
sage: omega = M.symplectic_form()
sage: X = M.vector_field_module().an_element()
sage: alpha = omega.flat(X)
sage: alpha.set_name('alpha')
sage: alpha.display()
alpha = 2 dq - 2 dp
sage: omega.sharp(alpha).display()
alpha_sharp = 2 e_q + 2 e_p

volume_form(contra=0)#

Liouville volume form \(\frac{1}{n!}\omega^n\) associated with the symplectic form \(\omega\), where \(2n\) is the dimension of the manifold.

INPUT:

contra – (default: 0) number of contravariant indices of the returned tensor

OUTPUT:

if contra = 0: volume form associated with the symplectic form
if contra = k, with \(1\leq k \leq n\), the tensor field of type (k,n-k) formed from \(\epsilon\) by raising the first k indices with the symplectic form (see method up())

EXAMPLES:

Volume form on \(\RR^4\):

sage: M = manifolds.StandardSymplecticSpace(4)
sage: omega = M.symplectic_form()
sage: vol = omega.volume_form() ; vol
4-form mu_omega on the Standard symplectic space R4
sage: vol.display()
mu_omega = dq1∧dp1∧dq2∧dp2

static wrap(form, name=None, latex_name=None)#

Define the symplectic form from a differential form.

INPUT:

form – differential \(2\)-form

EXAMPLES:

Volume form on the sphere as a symplectic form:

sage: from sage.manifolds.differentiable.symplectic_form import SymplecticForm
sage: M = manifolds.Sphere(2, coordinates='stereographic')
sage: vol_form = M.induced_metric().volume_form()                   # long time
sage: omega = SymplecticForm.wrap(vol_form, 'omega', r'\omega')     # long time
sage: omega.display()                                               # long time
omega = -4/(y1^4 + y2^4 + 2*(y1^2 + 1)*y2^2 + 2*y1^2 + 1) dy1∧dy2

class sage.manifolds.differentiable.symplectic_form.SymplecticFormParal(manifold: Union[VectorFieldModule, DifferentiableManifold], name: Optional[str], latex_name: Optional[str] = None)#

Bases: SymplecticForm, DiffFormParal

A symplectic form on a parallelizable manifold.

Note

If \(M\) is not parallelizable, the class SymplecticForm should be used instead.

INPUT:

manifold – module \(\mathfrak{X}(M)\) of vector fields on the manifold \(M\), or the manifold \(M\) itself
name – (default: omega) name given to the symplectic form
latex_name – (default: None) LaTeX symbol to denote the symplectic form; if None, it is formed from name

EXAMPLES:

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

sage: M.<q, p> = EuclideanSpace(name="R2", latex_name=r"\mathbb{R}^2")
sage: omega = M.symplectic_form(name='omega', latex_name=r'\omega')
sage: omega
Symplectic form omega on the Euclidean plane R2
sage: omega.set_comp()[1,2] = -1
sage: omega.display()
omega = -dq∧dp

poisson(expansion_symbol=None, order=1)#

Return the Poisson tensor associated with the symplectic form.

INPUT:

expansion_symbol – (default: None) symbolic variable; if specified, the inverse will be expanded in power series with respect to this variable (around its zero value)
order – integer (default: 1); the order of the expansion if expansion_symbol is not None; the order is defined as the degree of the polynomial representing the truncated power series in expansion_symbol; currently only first order inverse is supported

OUTPUT:

the Poisson tensor, , as an instance of PoissonTensorFieldParal()

EXAMPLES:

Poisson tensor of \(2\)-dimensional symplectic vector space:

sage: from sage.manifolds.differentiable.symplectic_form import SymplecticFormParal
sage: M.<q, p> = EuclideanSpace(2, "R2", r"\mathbb{R}^2", symbols=r"q:q p:p")
sage: omega = SymplecticFormParal(M, 'omega', r'\omega')
sage: omega[1,2] = -1
sage: poisson = omega.poisson(); poisson
2-vector field poisson_omega on the Euclidean plane R2
sage: poisson.display()
poisson_omega = -e_q∧e_p

restrict(subdomain, dest_map=None)#

Return the restriction of the symplectic form to some subdomain.

If the restriction has not been defined yet, it is constructed here.

INPUT:

subdomain – open subset \(U\) of the symplectic form’s domain
dest_map – (default: None) smooth destination map \(\Phi:\ U \rightarrow V\), where \(V\) is a subdomain of the symplectic form’s domain If None, the restriction of the initial vector field module is used.

OUTPUT:

the restricted symplectic form.

EXAMPLES:

Restriction of the standard symplectic form on \(\RR^2\) to the upper half plane:

sage: from sage.manifolds.differentiable.symplectic_form import SymplecticFormParal
sage: M = EuclideanSpace(2, "R2", r"\mathbb{R}^2", symbols=r"q:q p:p")
sage: X.<q, p> = M.chart()
sage: omega = SymplecticFormParal(M, 'omega', r'\omega')
sage: omega[1,2] = -1
sage: U = M.open_subset('U', coord_def={X: q>0})
sage: omegaU = omega.restrict(U); omegaU
Symplectic form omega on the Open subset U of the Euclidean plane R2
sage: omegaU.display()
omega = -dq∧dp