Poisson brackets coordinates

In order to construct the change of coordinates back to the original coordinates, we make use of the generating function W. Indeed we simply have to evaluate Poisson brackets but without solving any partial differential equations. Therefore the computational effort is much smaller than the one corresponding to the calculation of the normal-form Hamiltonian and the generating function. 

In terms of these results it can be shown that the value of a Poisson bracket is invariant under a canonical transformation of the coordinates. Like the corresponding commutator relationships in quantum mechanics, to which it is related by the expression... 

Of particular importance to the forthcoming development is the statement of the change induced in a property A(p, q) by the generator of an infinitesimal canonical transformation G as expressed in the Poisson bracket notation. The change in the property A obtained as a result of an infinitesimal canonical transformation of the coordinates is given by... 

In terms of the Poisson bracket, it is possible to write the differential equation corresponding to a coordinate g , say, as... 

Here a = WijdxiAdxj and sgrad / = ((sgrad /)i,..., (sgrad /)2n) Thus, the Poisson bracket of the functions f and g is a skew-symmetric scalar product (with respect to the form a ) of their skew gradients. If we denote by the coefficients of the matrix reciprocal to the matrix (c< y), then the Poisson bracket can be written in the local coordinates xi,..., X2n on Af as... 

For applications, it is instructive to know the explicit expression for the Poisson bracket of two functions in canonical symplectic coordinates. Let pi,..., be canonical coordinates in a symplectic space Then... 

The matrix U is therefore box-diagonal and its diagonal contains identical boxes coinciding with the matrix M, Hence rankL = rankL = AT rank Af = N(n — r). Since rank L is exactly the dimension of such an orbit of the coadjoint action which passes we arrive at the estimate indG Nr, For obtaining the inequality ind Ga Nr we suppose that on G there exist r functionally independent invariants Ji i = 1,. ..,r), that is, polynomial functions constant on the orbits. Then the extended functions (p = 1,.. j N) will be invariant functions on G. Indeed, invariance of any function JP is equivalent to the fact that it commutes in the sense of Poisson bracket with all coordinate functions J. We have... 

In this coordinate, the operation of the Poisson bracket with Flf becomes diagonal with respect to the monomials ... 

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