Momentum Poisson brackets

GENERIC tries to formulate a general time evolution equation by which the time evolution (derivative) of a state variable (which can be, e.g., mass density or fraction, momentum, energy) is determined by two potentials the total energy of the system and a dissipation function. Just the latter one introduces the irreversibility (and, in this way, the thermodynamics ) into consideration and description of the system behavior. The dissipation function or potential is a function of derivatives (with respect to the state variables) of a quantity which should have the physical meaning of the entropy of the system and this latter function is minimum at zero state variables, is zero at zero entropy derivatives just mentioned and a concave function. The general evolution equation can be reformulated by means of Poisson brackets. To apply the GENERIC formalism first one has to select suitable state variables for the problem or system which is to be modeled. The next step is to formulate... 

PROOF This assertion follows from 0-invariance of the momentum mapping and from the definition of the Poisson bracket. We shall dwell on this fact in more detail. In fact, for any functions a and P on G, we have the identity... 

A Lie algebra G is called compact if there exists a positive definite scalar product (, ) on G invariant under all inner automorphisms. Let a reductive Lie algebra G be a Lie algebra of functions with respect to the Poisson bracket on a symplectic manifold (A7,o ). The semisimplicity condition for the image of the momentum mapping F M G is automatically fulfilled by virtue of Theorem 3.3.6 for... 

This follows from the regularity theorem for analytic elliptic differential operators if F is a polynomial in momentum, then the condition that the Poisson bracket If, F) is identical zero determines the elliptic differential equation for the coefficients of the polynomial F,... 

Note that we have introduced the symbol 7t for linear momentum here in order to better distinguish it from canonical momentum. This notational rigor is only needed for the discussion of this section and will thus be dropped elsewhere in the book. In most cases it will become obvious from the context to which kind of momentum we are referring. Obviously, it is rather the linear than the canonical momentum which is gauge invariant. The canonical momentum p satisfies the fundamental Poisson brackets of Eq. (2.81), of course, whereas the components of linear momentum, interpreted as phase space functions tt, = Ki r,p,t), feature nonvanishing but gauge invariant Poisson brackets. 

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