The Jacobi bracket of a contact form #
Let α be a contact form on a finite-dimensional manifold M, with Reeb vector field R_α, and
write X_f for the contact Hamiltonian vector field of a function f, so that α(X_f) = f and
ι_(X_f) dα = (R_α f) α - df. The Jacobi bracket of two functions is
{f, g}_α = dα(X_f, X_g) + f R_α(g) - g R_α(f)
(TauCeti.SmoothForm.jacobiBracket). It is the bracket of functions induced by the Lie bracket of
contact vector fields: for C² functions,
{f, g}_α = α([X_f, X_g]) and X_({f, g}_α) = [X_f, X_g]
(TauCeti.SmoothForm.IsContactForm.apply_mlieBracket_contactHamiltonianVectorField and
TauCeti.SmoothForm.IsContactForm.contactHamiltonianVectorField_jacobiBracket). The second
identity rests on d(dα) = 0: the bracket of two contact Hamiltonian vector fields is again a
contact Hamiltonian vector field
(TauCeti.SmoothForm.IsContactHamiltonianVectorField.mlieBracket).
Consequently it is a Jacobi bracket in the sense of Lichnerowicz:
- it is antisymmetric (
TauCeti.SmoothForm.neg_jacobiBracket) and bilinear on differentiable functions (TauCeti.SmoothForm.jacobiBracket_add_left,TauCeti.SmoothForm.jacobiBracket_const_smul_left); - it is a first-order differential operator in each argument rather than a derivation, obeying
the generalized Leibniz rule
{f, g k} = g {f, k} + k {f, g} + g k R_α(f), whereR_α(f) = -{f, 1}(TauCeti.SmoothForm.IsContactForm.jacobiBracket_mul_right); - it satisfies the Jacobi identity
{f, {g, k}} + {g, {k, f}} + {k, {f, g}} = 0forC²functions (TauCeti.SmoothForm.IsContactForm.jacobiBracket_jacobi). The proof expandsX_({g, k}) = [X_g, X_k]and uses the Reeb derivative of the bracket,R_α({f, g}) = X_f(R_α g) - X_g(R_α f)(TauCeti.SmoothForm.IsContactForm.mvfderiv_jacobiBracket_reebVectorField).
Unlike a Poisson bracket, {1, g}_α = R_α(g) need not vanish
(TauCeti.SmoothForm.IsContactForm.jacobiBracket_const_left). The bracket can also be computed as
the derivative of g along X_f, corrected by the Reeb derivative of f:
{f, g}_α = X_f(g) - g R_α(f) (TauCeti.SmoothForm.IsContactForm.jacobiBracket_eq_mvfderiv).
References #
- Hansjörg Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics 109 (2008), Section 2.3.
- André Lichnerowicz, Les variétés de Jacobi et leurs algèbres de Lie associées, J. Math. Pures Appl. 57 (1978), 453–488.
The bracket #
The Jacobi bracket {f, g}_α = dα(X_f, X_g) + f R_α(g) - g R_α(f) of two functions, where
X_f and X_g are the contact Hamiltonian vector fields of f and g and R_α is the Reeb
vector field of α. For a contact form and C² functions it equals α([X_f, X_g])
(TauCeti.SmoothForm.IsContactForm.apply_mlieBracket_contactHamiltonianVectorField).
Instances For
The Jacobi bracket unfolds as dα(X_f, X_g) + f R_α(g) - g R_α(f).
The Jacobi bracket is antisymmetric.
The Jacobi bracket of a function with itself vanishes.
Negating the first argument negates the Jacobi bracket.
Negating the second argument negates the Jacobi bracket.
Multiplying the first argument by a real constant scales the Jacobi bracket.
Multiplying the second argument by a real constant scales the Jacobi bracket.
The Jacobi bracket is additive in its first argument on functions differentiable at the point.
The Jacobi bracket is additive in its second argument on functions differentiable at the point.
Contact Hamiltonian vector fields #
Derivative of a contact Hamiltonian along another contact Hamiltonian vector field. If X
and Y are contact Hamiltonian vector fields of f and g, then
X(g) = f R_α(g) + dα(X, Y).
The contact form on the bracket of contact Hamiltonian vector fields. If X and Y are
contact Hamiltonian vector fields of f and g, differentiable at x, then
α([X, Y]) = dα(X, Y) + f R_α(g) - g R_α(f) at x.
Contact forms #
The bracket of contact Hamiltonian vector fields is a contact Hamiltonian vector field.
For a contact form α and differentiable contact Hamiltonian vector fields X of f and Y of
g, with f and g of class C², the Lie bracket [X, Y] is the contact Hamiltonian vector
field of α([X, Y]).
The Jacobi bracket is the derivative of g along X_f, corrected by the Reeb derivative of
f: {f, g}_α = X_f(g) - g R_α(f).
The Jacobi bracket is minus the derivative of f along X_g, corrected by the Reeb
derivative of g: {f, g}_α = -X_g(f) + f R_α(g).
The Jacobi bracket with a constant c on the left is c times the Reeb derivative:
{c, g}_α = c R_α(g). In particular {1, g}_α = R_α(g), so the Jacobi bracket is not a Poisson
bracket.
The Jacobi bracket with a constant c on the right is -c times the Reeb derivative:
{f, c}_α = -c R_α(f).
The generalized Leibniz rule in the second argument, on functions differentiable at the
point: {f, g k}_α = g {f, k}_α + k {f, g}_α + g k R_α(f). Since R_α(f) = -{f, 1}_α, this is
the first-order Leibniz rule {f, g k} = g {f, k} + k {f, g} - g k {f, 1} of a Jacobi bracket.
The generalized Leibniz rule in the first argument, on functions differentiable at the
point: {g k, f}_α = g {k, f}_α + k {g, f}_α - g k R_α(f).
The Jacobi bracket is induced by the Lie bracket of contact vector fields:
α([X_f, X_g]) = {f, g}_α at points where X_f and X_g are differentiable, for instance
everywhere when f and g are C²
(TauCeti.SmoothForm.IsContactForm.mdifferentiable_contactHamiltonianVectorField).
The contact Hamiltonian vector field of a Jacobi bracket of C² functions is the Lie
bracket of their contact Hamiltonian vector fields: X_({f, g}_α) = [X_f, X_g].
The Reeb derivative of a Jacobi bracket. For C² functions f and g,
R_α({f, g}_α) = X_f(R_α g) - X_g(R_α f).
The Jacobi identity for the Jacobi bracket. For a contact form on a finite-dimensional
manifold and C² functions f, g, k,
{f, {g, k}_α}_α + {g, {k, f}_α}_α + {k, {f, g}_α}_α = 0.