Poisson tensors#

AUTHORS:

  • Tobias Diez (2021): initial version

class sage.manifolds.differentiable.poisson_tensor.PoissonTensorField(manifold: DifferentiableManifold | VectorFieldModule, name: str | None = 'varpi', latex_name: str | None = '\\varpi')[source]#

Bases: MultivectorField

A Poisson bivector field \(\varpi\) on a differentiable manifold.

That is, at each point \(m \in M\), \(\varpi_m\) is a bilinear map of the type:

\[\varpi_m:\ T^*_m M \times T^*_m M \to \RR\]

where \(T^*_m M\) stands for the cotangent space to the manifold \(M\) at the point \(m\), such that \(\varpi_m\) is skew-symmetric and the Schouten-Nijenhuis bracket (cf. bracket()) of \(\varpi\) with itself vanishes.

INPUT:

  • manifold – module \(\mathfrak{X}(M)\) of vector fields on the manifold \(M\), or the manifold \(M\) itself

  • name – (default: varpi) name given to the Poisson tensor

  • latex_name – (default: \\varpi) LaTeX symbol to denote the Poisson tensor; if None, it is formed from name

EXAMPLES:

A Poisson tensor 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: varpi = M.poisson_tensor(name='varpi', latex_name=r'\varpi')
sage: varpi
2-vector field varpi on the 2-sphere S^2 of radius 1 smoothly embedded
 in the Euclidean space E^3

>>> from sage.all import *
>>> M = manifolds.Sphere(Integer(2), coordinates='stereographic', names=('x', 'y',)); (x, y,) = M._first_ngens(2)
>>> stereoN = M.stereographic_coordinates(pole='north')
>>> stereoS = M.stereographic_coordinates(pole='south')
>>> varpi = M.poisson_tensor(name='varpi', latex_name=r'\varpi')
>>> varpi
2-vector field varpi on the 2-sphere S^2 of radius 1 smoothly embedded
 in the Euclidean space E^3

varpi 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: varpi[1, 2] = 1

>>> from sage.all import *
>>> varpi[Integer(1), Integer(2)] = Integer(1)

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

sage: varpi.add_comp_by_continuation(stereoS.frame(),
....:                  stereoS.domain().intersection(stereoN.domain()))
sage: varpi.display()
varpi = ∂/∂x∧∂/∂y
sage: varpi.display(stereoS)
varpi = (-xp^4 - 2*xp^2*yp^2 - yp^4) ∂/∂xp∧∂/∂yp

>>> from sage.all import *
>>> varpi.add_comp_by_continuation(stereoS.frame(),
...                  stereoS.domain().intersection(stereoN.domain()))
>>> varpi.display()
varpi = ∂/∂x∧∂/∂y
>>> varpi.display(stereoS)
varpi = (-xp^4 - 2*xp^2*yp^2 - yp^4) ∂/∂xp∧∂/∂yp

The Schouten-Nijenhuis bracket of a Poisson tensor with itself vanishes (this is trivial here, since M is 2-dimensional):

sage: varpi.bracket(varpi).display()
[varpi,varpi] = 0

>>> from sage.all import *
>>> varpi.bracket(varpi).display()
[varpi,varpi] = 0
hamiltonian_vector_field(function)[source]#

Return the Hamiltonian vector field \(X_f\) generated by the given function \(f: M \to \RR\).

The Hamiltonian vector field is defined by

\[X_f = - \varpi^\sharp (d f),\]

where \(\varpi^\sharp: T^* M \to TM\) is given by \(\beta(\varpi^\sharp(\alpha)) = \varpi(\alpha, \beta)\).

INPUT:

  • function – the function generating the Hamiltonian vector field

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(function('f')(q, p), name='f')
sage: Xf = poisson.hamiltonian_vector_field(f)
sage: Xf.display()
Xf = d(f)/dp e_q - d(f)/dq e_p

>>> from sage.all import *
>>> M = EuclideanSpace(Integer(2), names=('q', 'p',)); (q, p,) = M._first_ngens(2)
>>> poisson = M.poisson_tensor('varpi')
>>> poisson.set_comp()[Integer(1),Integer(2)] = -Integer(1)
>>> f = M.scalar_field(function('f')(q, p), name='f')
>>> Xf = poisson.hamiltonian_vector_field(f)
>>> Xf.display()
Xf = d(f)/dp e_q - d(f)/dq e_p
poisson_bracket(f, g)[source]#

Return the Poisson bracket

\[\{f, g\} = \varpi(df, dg)\]

of the given functions.

INPUT:

  • f – first function

  • g – second function

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(function('f')(q, p), name='f')
sage: g = M.scalar_field(function('g')(q, p), 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

>>> from sage.all import *
>>> M = EuclideanSpace(Integer(2), names=('q', 'p',)); (q, p,) = M._first_ngens(2)
>>> poisson = M.poisson_tensor('varpi')
>>> poisson.set_comp()[Integer(1),Integer(2)] = -Integer(1)
>>> f = M.scalar_field(function('f')(q, p), name='f')
>>> g = M.scalar_field(function('g')(q, p), name='g')
>>> 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
sharp(form)[source]#

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

\[\beta(\varpi^\sharp(\alpha)) = \varpi(\alpha, \beta).\]

for all \(\alpha, \beta \in T^*_m M\).

In indices, \(\alpha^i = \varpi^{ij} \alpha_j\).

INPUT:

  • form – the differential form to calculate its sharp of

EXAMPLES:

sage: M.<q, p> = EuclideanSpace(2)
sage: poisson = M.poisson_tensor('varpi')
sage: poisson.set_comp()[1,2] = -1
sage: a = M.one_form(1, 0, name='a')
sage: poisson.sharp(a).display()
a_sharp = e_p

>>> from sage.all import *
>>> M = EuclideanSpace(Integer(2), names=('q', 'p',)); (q, p,) = M._first_ngens(2)
>>> poisson = M.poisson_tensor('varpi')
>>> poisson.set_comp()[Integer(1),Integer(2)] = -Integer(1)
>>> a = M.one_form(Integer(1), Integer(0), name='a')
>>> poisson.sharp(a).display()
a_sharp = e_p
class sage.manifolds.differentiable.poisson_tensor.PoissonTensorFieldParal(manifold: DifferentiableManifold | VectorFieldModule, name: str | None = None, latex_name: str | None = None)[source]#

Bases: PoissonTensorField, MultivectorFieldParal

A Poisson bivector field \(\varpi\) on a parallelizable manifold.

INPUT:

  • manifold – module \(\mathfrak{X}(M)\) of vector fields on the manifold \(M\), or the manifold \(M\) itself

  • name – (default: varpi) name given to the Poisson tensor

  • latex_name – (default: \\varpi) LaTeX symbol to denote the Poisson tensor; if None, it is formed from name

EXAMPLES:

Standard Poisson tensor on \(\RR^2\):

sage: M.<q, p> = EuclideanSpace(2)
sage: varpi = M.poisson_tensor(name='varpi', latex_name=r'\varpi')
sage: varpi[1,2] = -1
sage: varpi
2-vector field varpi on the Euclidean plane E^2
sage: varpi.display()
varpi = -e_q∧e_p

>>> from sage.all import *
>>> M = EuclideanSpace(Integer(2), names=('q', 'p',)); (q, p,) = M._first_ngens(2)
>>> varpi = M.poisson_tensor(name='varpi', latex_name=r'\varpi')
>>> varpi[Integer(1),Integer(2)] = -Integer(1)
>>> varpi
2-vector field varpi on the Euclidean plane E^2
>>> varpi.display()
varpi = -e_q∧e_p