- Tautological one-form
In
mathematics , the tautological one-form is a special1-form defined on thecotangent bundle "T"*"Q" of amanifold "Q". Theexterior derivative of this form defines asymplectic form giving "T"*"Q" the structure of asymplectic manifold . The tautological one-form plays an important role in relating the formalism ofHamiltonian mechanics andLagrangian mechanics . The tautological one-form is sometimes also called the Liouville one-form, the canonical one-form, or the symplectic potential. A similar object is the canonicalvector field on thetangent bundle .In
canonical coordinates , the tautological one-form is given by:heta = sum_i p_i dq^i.,
Equivalently, any coordinates on phase space which preserve this structure for the canonical one-form, up to a total differential (
exact form ), may be called canonical coordinates; transformations between different canonical coordinate systems are known ascanonical transformation s.The canonical symplectic form is given by
:omega = -d heta = sum_i dq^i wedge 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 and M=T^*Q be thecotangent bundle orphase space . Let:pi:M o Q
be the canonical fiber bundle projection, and let
:T_pi:TM o 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 q=pi(m):
:m:T_qQ o mathbb{R}.
That is, we have that "m" is in the fiber of "q". The tautological one-form heta_m at point "m" is then defined to be
:heta_m = m circ T_pi
It is a linear map
:heta_m:T_mM o mathbb{R}
and so
:heta:TM o mathbb{R}.
Properties
The tautological one-form is the unique one-form that "cancels" a pullback. That is, let
:eta:Q o T^*Q
be any 1-form on "Q", and eta^* be its pullback. Then
:eta^* heta = eta
and
:eta^*omega = -deta
This can be most easily understood in terms of coordinates:
:eta^* heta = eta^*sum_i p_idq^i = sum_i eta^*p_idq^i = sum_i eta_idq^i = eta
Action
If "H" is a Hamiltonian on the
cotangent bundle and X_H is itsHamiltonian flow , then the corresponding action "S" is given by:S= heta (X_H).
In more prosaic terms, the Hamiltonian flow represents the classical trajectory of a mechanical system obeying the
Hamilton-Jacobi equations of motion . The Hamiltonian flow is the integral of the Hamiltonian vector field, and so one writes, using traditional notation foraction-angle variables ::S(E) = sum_i oint p_i,dq^i
with the integral understood to be taken over the manifold defined by holding the energy E constant: H=E=const. .
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 map:g:TQ o T^*Q,
then define
:Theta = g^* heta
and
:Omega = -dTheta = g^*omega
In generalized coordinates q^1,ldots,q^n,dot q^1,ldots,dot q^n) on "TQ", one has
:Theta=sum_{ij} g_{ij} dot q^i dq^j
and
:Omega= sum_{ij} g_{ij} ; dq^i wedge ddot q^j +sum_{ijk} frac{partial g_{ij{partial q^k} ; dot q^i, dq^j wedge dq^k
The metric allows one to define a unit-radius sphere in T^*Q. The canonical one-form restricted to this sphere forms a
contact structure ; the contact structure may be used to generate thegeodesic flow for this metric.ee also
*
fundamental class References
*
Ralph Abraham and Jarrold E. Marsden, "Foundations of Mechanics", (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X "See section 3.2".
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