Co-tangent bundle

This is a system of linear equations for A. Assuming g ( )Af g ( ) is invertible, we may solve for A. Reinserting this expression for A into (4.8) then gives a system of ordinary differential equations whose solutions, starting from a point on T M, will remain on T M for all time. The co-tangent bundle becomes an invariant manifold of this ODE system. A is defined at any time by the physical variables q andp. 

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. 

Thus we see that the projected version of the symplectic form in the ambient space is conserved by the differential equation system (4.7)-(4.9). This is what is meant by saying that the constrained system is analogous to a Hamiltonian system. We may also think of its flow map as being a symplectic map of the co-tangent bundle. 

It is difficult to make sense of this question, since (4.14)-(4.16) does not even constitute a map of the co-tangent bundle. On the other hand it is not too difficult to correct this defect by incorporating an additional projection onto the cotangent space ... 

For Symplectic Euler with constraints, we view Qh as a mapping of the co-tangent bundle and write Q,P) = Qh(s,p) where... 

Lemma 1.3.1. The level surface M23 is diffeomorpbic to the (co)tangent bundle of a two-dimensional sphere. 

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