# Dirac bracket on a Poisson manifold in relation to the Courant bracket on the Whitney sum.

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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!

asked Jun 6, 2015

Your question asks too much at the same time. This is an extensive subject and hardly anybody likes to write an article-sized answer. It takes time to understand how different formalisms are related - trying to understand them all at once is quite confusing. The usual thing is that one begins by understanding well one of the approaches, then the next one, etc., starting with the one that looks easiest or most promising.

Maybe you can tell us what your real goals are - how would you use the information that you asked about?

@ArnoldNeumaier but it might be possible that somebody who has not only a deep knowledge but also a broad overview about the topic is possible to give an answer, that some kind of reviews the different formalisms.

@dilaton - Even if that were the case it is better to have shorter, focused questions, if needed several. It is less frustrating to answer and it makes for better answers

In any case, at present we haven't anyone of this sort around and it is important to constrain the question. Answering it as it is now would require me to first make an extended literature search....

@ArnoldNeumaier ok

I just thought that maybe @UrsSchreiber could have something something to say about it?

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