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Poisson bracket relations for generators of canonical transformations are derived directly from the Galilei and Poincare groups of changes of space-time coordinates. The method is simple but rigorous. The meaning of each step is clear because it corresponds to an operation in the group of changes of space-time coordinates. Only products and inverses are used; differences are not used. It is made explicitly clear why constants occur in some bracket relations but not in others, and how some constants can be removed, so that in the end there is a constant in the bracket relations for the Galilei group but not for the Poincare group. Each change of coordinates needs to be only to first order, so matrices are not needed for rotations or Lorentz transformations; simple three-vector descriptions are enough. Conversion to quantum mechanics is immediate. One result is a simpler derivation of the commutation relations for angular momentum directly from rotations. Problems are included.
We analyze the transformation properties of Faraday law in an empty space and its relationship with Maxwell equations. In our analysis we express the Faraday law via the four-potential of electromagnetic field and the field of four-velocity, defined
A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $mathrm U_q(mathcal L(mathfrak{sl}_3))$ is given. The full proof of the functional relations in the form independent of
We report Relativity tests based on data from two simultaneous Michelson-Morley experiments, spanning a period of more than one year. Both were actively rotated on turntables. One (in Berlin, Germany) uses optical Fabry-Perot resonators made of fused
We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classi
The Dubrovin-Zhang hierarchy is a Hamiltonian infinite-dimensional integrable system associated to a semi-simple cohomological field theory or, alternatively, to a semi-simple Dubrovin-Frobenius manifold. Under an extra assumption of homogeneity, Dub