Symplectic Structure with Holonomic Constraints

Describing the Hamiltonian structure for a constrained system is a little complicated to do formally. The simplifying concept that we exploit is that the symplectic 2-form in the ambient space can be projected to the co-tangent bundle to define an associated symplectic form on the manifold. 

Taking the time derivative, as the right hand side is constant we have 

Taking the derivative with respect to t and defining Z,- =fi Z(t,z)), we can see that 

We next demonstrate that a similar property holds for the constrained equations. Taking differentials, we have 

we interchange time derivatives and differentials, and use the chain rule to write 

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