The Liouville form and the canonical symplectic form of a cotangent bundle #
The cotangent bundle T*M = TauCeti.CotangentBundle I M of a manifold carries a tautological, or
Liouville, one-form λ: at a covector p over x, it evaluates a tangent vector v of T*M by
λ_p(v) = p(dπ v),
where π : T*M → M is the bundle projection (TauCeti.CotangentBundle.liouvilleForm). Its
negative exterior derivative is the canonical symplectic form ω = -dλ
(TauCeti.CotangentBundle.symplecticForm).
Both forms are computed in the preferred charts of T*M, with coordinates (q, p): there λ is
the linear Liouville form λ_(q,p)(δq, δp) = p(δq), that is Σ pᵢ dqᵢ
(TauCeti.CotangentBundle.inChartAt_liouvilleForm), and ω is the linear form
TauCeti.strongDualCotangentSymplecticForm, that is Σ dqᵢ ∧ dpᵢ
(TauCeti.CotangentBundle.inChartAt_symplecticForm_apply). The chart formula for λ reduces to
the derivative of the projection in a bundle chart,
Bundle.mfderiv_proj_comp_mfderivWithin_extChartAt_symm; it gives smoothness of λ, and that
of ω follows from the linear identity TauCeti.neg_extDeriv_cotangentLiouvilleForm. The sign
conventions are therefore those of the linear model. Consequently ω is closed and nondegenerate
(TauCeti.CotangentBundle.isSymplectic_symplecticForm).
The form λ is tautological: its pullback along the section of T*M defined by a one-form α
is α itself (TauCeti.CotangentBundle.mpullback_liouvilleForm, and its pointwise version
TauCeti.CotangentBundle.mpullback_liouvilleForm_section). The base may have boundary or corners,
and nothing here assumes finite dimension.
Main declarations #
TauCeti.CotangentBundle.liouvilleForm: the Liouville one-form onT*M.TauCeti.CotangentBundle.inChartAt_liouvilleForm: its chart formulaΣ pᵢ dqᵢ.TauCeti.CotangentBundle.mpullback_liouvilleForm: its pullback by the section of a one-form.TauCeti.CotangentBundle.symplecticForm: the canonical symplectic formω = -dλ.TauCeti.CotangentBundle.inChartAt_symplecticForm_apply: its chart formula.TauCeti.CotangentBundle.isSymplectic_symplecticForm:ωis symplectic.
References #
- A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001, Section 2.
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 3.2.
The Liouville form (tautological one-form) of the cotangent bundle T*M: at a covector
p over x, it evaluates a tangent vector v of T*M by p (dπ v), where π : T*M → M is the
bundle projection. It is a smooth one-form on T*M, and in every bundle chart it is the linear
Liouville form TauCeti.cotangentLiouvilleForm, that is Σ pᵢ dqᵢ.
Instances For
The Liouville form evaluates a tangent vector of T*M at the covector z by applying z to
the image of the vector under the derivative of the bundle projection.
Chart formula for the Liouville form. In every preferred chart of the cotangent bundle,
with coordinates (q, p), the Liouville form is the linear Liouville form
λ_(q,p)(δq, δp) = p(δq), that is Σ pᵢ dqᵢ.
The Liouville form pulled back along a map ψ : X → T*M lying over a map g : X → M
evaluates a tangent vector v at x by applying the covector ψ x to dg v, provided ψ is
differentiable at x.
The Liouville form is tautological. Pulling it back along the section x ↦ σ x of the
cotangent bundle defined by a family of covectors σ gives back σ, at every point where that
section is differentiable.
The Liouville form is tautological. Pulling it back along the smooth section of the
cotangent bundle defined by a smooth one-form α gives back α.
The Liouville form pulled back by a smooth map ψ : T*M → T*N covering a map f : M → N
evaluates a tangent vector at z by applying the covector ψ z to df (dπ v). The base map f
is automatically smooth, as it is x ↦ π (ψ (0_x)).
The Liouville form pulled back by a smooth self-map g of the cotangent bundle covering the
identity evaluates a tangent vector at z by applying the covector g z to its image under the
derivative of the projection. This is the f = id case of
mpullback_liouvilleForm_apply_of_proj_apply_eq.
Rescaling covectors rescales the pulled-back Liouville form. If a map φ into the
cotangent bundle multiplies each covector ψ y of a map ψ by a scalar c y, keeping its base
point, then φ^*λ = c • ψ^*λ. The scalar function c need not be smooth or nonzero.
The canonical symplectic form ω = -dλ of the cotangent bundle T*M, where λ is the
Liouville form TauCeti.CotangentBundle.liouvilleForm. In every bundle chart it is the linear
form TauCeti.strongDualCotangentSymplecticForm, that is Σ dqᵢ ∧ dpᵢ.
Instances For
The canonical symplectic form is minus the bundled exterior derivative of the Liouville form.
The canonical symplectic form is minus the exterior derivative of the Liouville form.
Chart formula for the canonical symplectic form. In every preferred chart of the cotangent
bundle, the canonical symplectic form is the linear form
TauCeti.strongDualCotangentSymplecticForm, ω((δq, δp), (δq', δp')) = δp'(δq) - δp(δq').
At a covector z, read in the preferred chart at z itself, where tangent vectors are their
own coordinates, the canonical symplectic form is the linear form.
The canonical symplectic form, evaluated on an ordered pair of tangent vectors at a covector
z, is the linear form TauCeti.strongDualCotangentSymplecticForm.
The canonical symplectic form of the cotangent bundle is closed and nondegenerate.