Geometry from Dynamics, Classical and Quantum by José F. Cariñena Alberto Ibort Giuseppe Marmo & Giuseppe Morandi

Author:José F. Cariñena, Alberto Ibort, Giuseppe Marmo & Giuseppe Morandi
Language: eng
Format: epub
Publisher: Springer Netherlands, Dordrecht

(5.158)

and, as a consequence of ,

(5.159)

i.e., will vanish on any pair of vertical fields.

Recall that is a derivation of degree zero such that

Noticing that and , this last result can be rewritten as

(5.160)

5.5.3.2 Conditions Under Which a Closed 2-Form Is a Lagrangian 2-Form

We have seen in the previous Subsection that to every function (let it be regular or not) we can associate (a Cartan 1-form and) a Lagrangian 2-form which is closed (actually exact) and vanishes on pairs of vertical fields. A relevant question is then under which conditions such a 2-form on is actually a 2-form for some . This is settled by the following [Ca95]:

Theorem 5.39

Let be a closed 2-form, . Then, will be a Lagrangian 2-form, , for some iff

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