Tautological one-form
In Mathematics, the tautological one-form is a special 1-form defined on symplectic manifolds that plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. In Local coordinates, the canonical symplectic form is exact; the tautological one-form is the one-form whose differential is (minus) the symplectic form on the symplectic manifold. The tautological one-form is sometimes also called the canonical one-form or the symplectic potential.
In Canonical coordinates, the tautological one-form is given by
θ =
Σ
ip i dq i
The canonical symplectic form is then given by
ω = -d θ =
Σ
idq i ∧ dp i
Coordinate-free definition
The tautological 1-form can also be defined rather abstractly as a form on phase space. Let Q be a manifold andπ : M → Q
be the canonical fiber bundle projection, and let
T π : TM → TQ
be the induced pushforward. Let m be a point on M, however, since M is the cotangent bundle, we can understand m to be a map of the tangent space at
m : T q Q → R .
That is, we have that m is in the fiber of q. The tautological one-form
θ m = m ∘ T π
It is a linear map
θ m : T m M → R
and so
θ : TM → R .
Properties
The tautological one-form is the unique one-form that "cancels" a Pullback. That is, letβ : Q → T * Q
be any 1-form on Q, and
β * θ = β
and
β * ω = -d β
This can be most easily understood in terms of coordinates:
β * θ = β *
Σ
ip i dq i = n
Σ
iβ * p i dq i =
Σ
iβ i dq i = β
Action
If H is a Hamiltonian on the Cotangent bundle andS = θ (X H ) .
On metric spaces
If the manifold Q has a Riemannian or pseudo-Riemannian Metric g, then corresponding definitions can be made in terms of generalized coordinates. Specifically, if we take the metric to be a mapg : TQ → T * Q ,
then define
Θ = g * θ
and
Ω = -d Θ = g * ω
In generalized coordinates ( q 1 , …,q n , .
q
1 , …, .
q
n )
Θ =
Σ
ijg ij .
q
i dq j
and
Ω =
Σ
ijg ij dq i ∧ d .
q
j + n
Σ
ijk∂ g ij
–––––––––
∂ q kn .
q
i dq j ∧ dq k