In classical dynamics we suppose that constants of the motion act like constraints reducing the phase space by one dimension to a hypersurface for each conserved coordinate. The intersection of these is the region of phase space the system is "allowed on". These can be considered as symplectic leaves in a Poisson manifold (please correct me if this is wrong).
When the constraints are not Casimir functions, rather they are imposed on the symplectic space itself then we need to satisfy them using the Dirac bracket if they are secondary constraints. The Dirac bracket is a tool that searches for the submanifold which satisfies the constraints. Upon this sub manifold the Dirac bracket reduces to the Poisson bracket. The Dirac bracket is a bracket on this sub manifold written in terms of Poisson brackets in the Poisson manifold. This is my understanding of the Dirac bracket.
The Poisson manifold can be generalised to the idea of the Whitney sum of tangent and cotangent bundles $TM\bigoplus T^*M$ which is useful for constrained dynamics. This space is endowed with the Courant bracket that generates a Courant algebroid structure. When the Courant bracket is confined to the submanifold it becomes the Dirac bracket.
My questions are;
0) Guidance on the above information!
1) How does the above interplay with the Dirac manifold we hear so much about in constrained dynamics?
2) How is this different from the ideas of Skinner and Rusk who use a Whitney sum to describe dynamics too!
3) Are these part of the same theory of constrained dynamics or are they all different things (this is something I'm struggling with). "These" includes: Skinner-Rusk formalism, Courant brackets, Dirac brackets, Nambu brackets (I have been told is a constrained dynamics formalism), Gotay-Nester algorithms, presymplectic geometry ...
I understand this is a very broad question, but help on any of the areas is more than appreciated! Please see my other question on the other physics forum. Many thanks!