Hamiltonian vector fields on symplectic manifolds #
Let σ be a smooth two-form on a manifold M. A vector field X is a Hamiltonian vector field
of a function f : M → ℝ when ι_X σ = df, that is σ(X x, v) = df_x(v) for every tangent
vector v at every point x (TauCeti.SmoothTwoForm.IsHamiltonianVectorField). This is the
sign convention of the vector-space API TauCeti.SymplecticForm.hamiltonianVectorField.
The Hamiltonian vector field of f is X_f = (σ♭)⁻¹ df, the inverse flat map
TauCeti.SmoothTwoForm.flat applied to the differential of f
(TauCeti.SmoothTwoForm.hamiltonianVectorField). On a finite-dimensional manifold every flat map
of a nondegenerate form is invertible, so X_f is then the unique Hamiltonian vector field of f
(TauCeti.SmoothTwoForm.IsNondegenerate.isHamiltonianVectorField_iff_eq). Smoothness of the
inverse flat maps makes the Hamiltonian vector field of a C^(n+1) function a C^n vector field
(TauCeti.SmoothTwoForm.IsNondegenerate.contMDiff_hamiltonianVectorField), which is the regularity
needed to integrate it to a Hamiltonian flow. Rescaling the form by c ≠ 0 rescales the
Hamiltonian vector field by c⁻¹
(TauCeti.SmoothTwoForm.IsNondegenerate.hamiltonianVectorField_smul), and post-composing the
Hamiltonian with a real function φ rescales it pointwise by φ', X_{φ ∘ f} = φ'(f) X_f
(TauCeti.SmoothTwoForm.hamiltonianVectorField_comp). The value of X_f at x only depends on the
germ of f at x (Filter.EventuallyEq.hamiltonianVectorField_eq).
Since σ is alternating, df(X) = σ(X, X) = 0 for a Hamiltonian vector field X of f: the
Hamiltonian is constant along the integral curves of X, which is conservation of energy
(TauCeti.SmoothTwoForm.IsHamiltonianVectorField.apply_eq_of_isMIntegralCurveOn). The orbits of
G + f ∘ F, where G is supported in {F < s} and f is constant on (-∞, s], either stay in
{F < s} as orbits of G or lie in a level of F as orbits of f'(c) X_F
(TauCeti.SmoothTwoForm.IsNondegenerate.isMIntegralCurve_or_exists_of_isMIntegralCurve_add_comp).
For a constant symplectic form on a vector space, the construction recovers the Hamiltonian vector
field of the linear theory (TauCeti.SymplecticForm.hamiltonianVectorField_constSmooth).
The Poisson bracket of two functions is {f, g} = σ(X_f, X_g)
(TauCeti.SmoothTwoForm.poissonBracket), which is df(X_g) for a nondegenerate form on a
finite-dimensional manifold. It is antisymmetric and agrees with the Poisson bracket of the linear
theory for a constant symplectic form (TauCeti.SymplecticForm.poissonBracket_constSmooth).
Hamiltonian vector fields pull back: if a smooth map φ carries a vector field X to a
Hamiltonian vector field of g, then X is a Hamiltonian vector field of g ∘ φ for the pulled
back form (TauCeti.SmoothTwoForm.IsHamiltonianVectorField.mpullback). On a product M × N with
the product form, the product of Hamiltonian vector fields of f and g is a Hamiltonian vector
field of (x, y) ↦ f x + g y (TauCeti.SmoothTwoForm.IsHamiltonianVectorField.prod).
Both constructions are natural. Pointwise, if a surjective linear map D from T_x M to
T_y M₂ satisfies σ₂_y(D u, D v) = σ_x(u, v) and dg_y ∘ D = df_x for nondegenerate σ and
σ₂, then D carries X_f(x) to X_g(y)
(TauCeti.SmoothTwoForm.IsNondegenerate.map_hamiltonianVectorField_of_eval₂). Surjectivity is
automatic when M₂ = M and σ₂ = σ
(TauCeti.SmoothTwoForm.IsNondegenerate.surjective_of_eval₂_map). Hence a smooth map φ pulling
a nondegenerate form σ₂ back to σ
carries the Hamiltonian vector field of g ∘ φ to that of g, wherever its differential is
surjective (TauCeti.SmoothTwoForm.IsNondegenerate.mfderiv_hamiltonianVectorField_comp), and so
{g ∘ φ, k ∘ φ} = {g, k} ∘ φ there (TauCeti.SmoothTwoForm.IsNondegenerate.poissonBracket_comp).
Along a curve where its differential is surjective, φ therefore maps integral curves of the
Hamiltonian vector field of g ∘ φ to integral curves of that of g
(TauCeti.SmoothTwoForm.IsNondegenerate.isMIntegralCurve_comp_hamiltonianVectorField).
Main declarations #
TauCeti.SmoothTwoForm.IsHamiltonianVectorField: the equationι_X σ = df, also in the language of forms (TauCeti.SmoothTwoForm.isHamiltonianVectorField_iff_interior_eq).TauCeti.SmoothTwoForm.hamiltonianVectorField: the Hamiltonian vector field(σ♭)⁻¹ df.TauCeti.SmoothTwoForm.IsNondegenerate.isHamiltonianVectorField_iff_eq: existence and uniqueness of the Hamiltonian vector field.TauCeti.SmoothTwoForm.IsNondegenerate.contMDiff_hamiltonianVectorField: aC^(n+1)Hamiltonian has aC^nHamiltonian vector field, with the versionTauCeti.SmoothTwoForm.IsNondegenerate.contMDiff_hamiltonianVectorField_uncurryfor jointlyC^(n+1)families of Hamiltonians, such as time-dependent ones.TauCeti.SmoothTwoForm.IsNondegenerate.hamiltonianVectorField_smul:X^{cσ}_f = c⁻¹ X^σ_f.TauCeti.SmoothTwoForm.IsHamiltonianVectorField.apply_eq_of_isMIntegralCurveOn: conservation of energy along integral curves on preconnected sets of times, with the global caseTauCeti.SmoothTwoForm.IsHamiltonianVectorField.apply_eq_of_isMIntegralCurve.TauCeti.SmoothTwoForm.IsNondegenerate.isMIntegralCurve_or_exists_of_isMIntegralCurve_add_comp: the dichotomy for the integral curves ofX_{G + f ∘ F}.TauCeti.SmoothTwoForm.IsHamiltonianVectorField.mpullback,TauCeti.SmoothTwoForm.IsHamiltonianVectorField.prod: Hamiltonian vector fields under pullback along related fields and on products.TauCeti.SymplecticForm.hamiltonianVectorField_constSmooth: comparison with the vector-space construction.TauCeti.SmoothTwoForm.poissonBracket: the Poisson bracket{f, g} = σ(X_f, X_g), withTauCeti.SmoothTwoForm.IsNondegenerate.poissonBracket_eq_mvfderiv:{f, g} = df(X_g), andTauCeti.SmoothTwoForm.IsNondegenerate.contMDiff_poissonBracket: the Poisson bracket ofC^(n+1)functions isC^n.TauCeti.SmoothTwoForm.IsNondegenerate.map_hamiltonianVectorField_of_eval₂: a surjective linear mapDwithσ₂(D u, D v) = σ(u, v)anddg ∘ D = dfcarriesX_f(x)toX_g(y).TauCeti.SmoothTwoForm.IsNondegenerate.mfderiv_hamiltonianVectorField_comp,TauCeti.SmoothTwoForm.IsNondegenerate.isMIntegralCurve_comp_hamiltonianVectorFieldandTauCeti.SmoothTwoForm.IsNondegenerate.poissonBracket_comp: naturality of Hamiltonian vector fields, their integral curves and Poisson brackets under maps pulling one form back to the other.
References #
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford, 2017, Section 3.1 (Hamiltonian vector fields on symplectic manifolds).
- H. Hofer, E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics, Birkhäuser, 1994, Chapter 1.
Hamiltonian vector fields #
A vector field X is a Hamiltonian vector field of f for the two-form σ when
ι_X σ = df: σ(X x, v) = df_x(v) for every tangent vector v at every point x.
Instances For
The defining characterization of a Hamiltonian vector field.
A Hamiltonian vector field is one whose flat is the differential of the Hamiltonian.
A Hamiltonian vector field in the language of forms: the interior product ι_X σ is the
exterior derivative of the 0-form of the Hamiltonian.
A Hamiltonian is stationary along its Hamiltonian vector field: df(X) = σ(X, X) = 0.
Conservation of energy. A Hamiltonian, differentiable along an integral curve of its Hamiltonian vector field on an interval of times (or any preconnected set of times), is constant along that curve.
Conservation of energy along a global integral curve: a Hamiltonian, differentiable along an integral curve of its Hamiltonian vector field, is constant along that curve.
The Hamiltonian vector field X_f = (σ♭)⁻¹ df of a function f. It satisfies
ι_{X_f} σ = df when σ is nondegenerate and M is finite-dimensional
(TauCeti.SmoothTwoForm.IsNondegenerate.isHamiltonianVectorField_hamiltonianVectorField); the
inverse is ContinuousLinearMap.inverse, so the field vanishes at points where the flat map is
not invertible.
Instances For
The Hamiltonian vector field unfolds as the inverse flat map applied to df.
The Hamiltonian vector field is local: at x it depends only on the germ of the
Hamiltonian at x.
The Hamiltonian vector field of a constant function vanishes.
The Hamiltonian vector field vanishes at every point near which the Hamiltonian is constant, for instance outside the topological support of the Hamiltonian.
The Hamiltonian vector field is additive on differentiable functions.
The Hamiltonian vector field of -f is the opposite of that of f.
The Hamiltonian vector field commutes with multiplication by a real constant.
Adding a real multiple of a differentiable Hamiltonian changes its vector field affinely.
At fixed x, the derivative in the real coefficient of f + t • g, read in the model
space of the tangent fiber, is the Hamiltonian vector of g at x.
The chain rule for Hamiltonian vector fields. Post-composing the Hamiltonian with a real
function φ differentiable at f x rescales the Hamiltonian vector field at x by φ'(f x):
X_{φ ∘ f}(x) = φ'(f x) X_f(x).
The flat of the Hamiltonian vector field is the differential of the Hamiltonian.
The zeros of a Hamiltonian vector field are the critical points of the Hamiltonian:
X_f(x) = 0 exactly when df_x = 0.
Contraction with the Hamiltonian vector field is the differential of the Hamiltonian,
σ(X_f, v) = df(v).
The simp normal form of eval₂_hamiltonianVectorField, with eval₂ unfolded:
σ_x(X_f, v) = df_x(v).
The Hamiltonian vector field of f is a Hamiltonian vector field of f.
Existence and uniqueness of Hamiltonian vector fields. For a nondegenerate two-form on a
finite-dimensional manifold, X_f is the only Hamiltonian vector field of f.
Regularity of Hamiltonian vector fields. For a nondegenerate two-form on a
finite-dimensional manifold, the Hamiltonian vector field of a C^(n+1) function is a C^n
vector field, for n ≤ ∞.
Joint regularity of Hamiltonian vector fields of a family. For a nondegenerate two-form on
a finite-dimensional manifold and a family of Hamiltonians H : N → M → ℝ which is jointly
C^(n+1) on N × M, the Hamiltonian vector fields (p, x) ↦ X_{H p}(x) are jointly C^n, for
n ≤ ∞. For N = ℝ this is the regularity of the vector field of a time-dependent Hamiltonian,
which is what is needed to integrate it to a time-dependent Hamiltonian flow.
The Hamiltonian vector field of a C² function is differentiable.
Hamiltonian vector fields of a rescaled form. For a nondegenerate form σ and c ≠ 0,
X^{cσ}_f = c⁻¹ X^σ_f.
Rescaling the form and the Hamiltonian by the same c ≠ 0 leaves the Hamiltonian vector
field unchanged: X^{cσ}_{cf} = X^σ_f.
Orbits of a Hamiltonian enlarged along the levels of F. Let G be supported in
{F < s}, let f be constant on (-∞, s], and let γ be an integral curve of the Hamiltonian
vector field of G + f ∘ F. Either γ stays in {F < s} and is an integral curve of X_G, or it
lies in a level F⁻¹(c) with c ≥ s, along which X_G vanishes, and is an integral curve of
f'(c) X_F.
The Poisson bracket #
The Poisson bracket {f, g}(x) = σ(X_f x, X_g x) of two functions. For a nondegenerate
form on a finite-dimensional manifold it is {f, g} = df(X_g)
(TauCeti.SmoothTwoForm.IsNondegenerate.poissonBracket_eq_mvfderiv).
Instances For
The Poisson bracket unfolds as σ evaluated on the two Hamiltonian vector fields.
The Poisson bracket is antisymmetric.
The Poisson bracket is additive in its first argument on differentiable functions.
The Poisson bracket is additive in its second argument on differentiable functions.
Negating the first argument negates the Poisson bracket.
Negating the second argument negates the Poisson bracket.
Multiplication of the first argument by a real constant scales the Poisson bracket.
Multiplication of the second argument by a real constant scales the Poisson bracket.
The Poisson bracket {f, g} is the derivative of f along the Hamiltonian vector field of
g.
The Poisson bracket {f, g} is minus the derivative of g along the Hamiltonian vector
field of f.
Regularity of the Poisson bracket. For a nondegenerate two-form on a finite-dimensional
manifold, the Poisson bracket of two C^(n+1) functions is C^n, for n ≤ ∞.
Pullbacks and products #
Hamiltonian vector fields pull back along related fields. Let Y be a Hamiltonian vector
field of g for σ₂ and let a smooth map φ carry a vector field X to Y, that is
dφ (X x) = Y (φ x). Then X is a Hamiltonian vector field of g ∘ φ for the pullback φ^* σ₂.
No nondegeneracy is needed.
Hamiltonian vector fields of products. If X is a Hamiltonian vector field of f for
σ on M and Y one of g for σ₂ on M₂, then the product field (X, Y) is a Hamiltonian
vector field of (x, y) ↦ f x + g y for the product form pr₁^* σ + pr₂^* σ₂ on M × M₂.
Naturality under symplectic maps #
Linear maps preserving nondegenerate forms carry Hamiltonian vectors. Let D be a
surjective linear map from the tangent space of M at x to the tangent space of M₂ at y
with σ₂_y(D u, D v) = σ_x(u, v) and dg_y ∘ D = df_x. Then D carries X_f(x) to X_g(y).
Hamiltonian vector fields are natural. Let a smooth map φ pull a nondegenerate form σ₂
back to a nondegenerate form σ. At a point where the differential of φ is surjective, it carries
the Hamiltonian vector field of g ∘ φ to the Hamiltonian vector field of g.
Poisson brackets are natural. Let a smooth map φ pull a nondegenerate form σ₂ back to a
nondegenerate form σ. At a point where the differential of φ is surjective, the Poisson
bracket of g ∘ φ and k ∘ φ is the Poisson bracket of g and k at the image point.
Hamiltonian integral curves are natural. Let a smooth map φ pull a nondegenerate form
σ₂ back to a nondegenerate form σ. If γ is an integral curve of the Hamiltonian vector field
of g ∘ φ, along which the differential of φ is surjective and g is differentiable at the
image points, then φ ∘ γ is an integral curve of the Hamiltonian vector field of g.
For a constant symplectic form on a vector space, the Hamiltonian vector field of the manifold theory is that of the linear theory.
For a constant symplectic form on a vector space, the Poisson bracket of the manifold theory is that of the linear theory.