Mass in Symplectic techniques in physics by Guillemin and Sternberg

[mma]

Mass in "Symplectic techniques in physics" by Guillemin and Sternberg

Hi,

have somebody read this book? Guillemin and Sternberg introduces the notion of mass on page 435. But I don't understand here everything. For example, I dont't know what is $$\sigma$$ (it is introduced but isn't used) and what is d₀. The $$\Psi$$ is also unclear, once it is a map from M to Z²(g) another time it is a cocycle. Could somebody help me to understand this? I haven't found elsewhere the same treatment and I find it very interesting, so I'd like to understand it.

 

[eljose79]

Hello,

I have read "Symplectic techniques in physics" by Guillemin and Sternberg and I can try to help clarify some of the concepts you mentioned. The \sigma that is introduced is the symplectic form, which is a fundamental concept in symplectic geometry. It is a non-degenerate, closed 2-form on a symplectic manifold that encodes the geometric structure of the manifold. The d0 is the exterior derivative of the symplectic form, which is used in the definition of the Hamiltonian vector field.

The \Psi function is indeed a map from the symplectic manifold M to the group of symplectomorphisms of the symplectic vector space Z2(g). This means that it is a map that preserves the symplectic structure of the manifold. However, in the context of mass, \Psi is used as a cocycle, which is a function that assigns a value to each element in a group and satisfies certain conditions. In this case, \Psi is used to define the mass cocycle, which is a fundamental concept in symplectic mechanics.

I hope this helps clarify some of the concepts for you. If you have any further questions, please feel free to ask. The treatment of mass in "Symplectic techniques in physics" is indeed very interesting and I'm glad you are interested in understanding it. Keep exploring and learning!