Mathematik und Theoretische Physik, I

Why must we work in the phase space?

“…This very special role of "microcanonical" distributions of A and their obvious property (35) fix our attention on the status of the joint condition (36) in general. Apparently the two conditions in (36) when taken separately are of a qualitatively different nature. The first of them, induced by the associative algebraic structure in C ∞ (P ), expresses the statistical-informational properties of ̺ -the lack of spread in the set of outcomes of A-measurements.…”

“…(ii) If integral curves of the Hamiltonian vector field X A are dense in valuesurfaces M (A,a) , i.e., if the corresponding dynamical systems on M (A,a) are "ergodic", then F |M (A,a) is constant, and δ(A − a) (up to a constant multiplier) is the only solution of (36). The reason is that the only globally defined, one-valued and smooth "constants of motion" have the form…”

“…{A, ̺} = 0, i.e., the second subcondition of (36). This is to be understood weakly (in the Dirac sense),…”

“…The situation becomes complicated when L is not invertible. Then according to the Dirac procedure [24]- [27], [36], [87], a technically difficult problem of dynamical constraints appears. But even in the regular case the "impetus" approach based on T * Q is much more adequate, just as Buridan and Olvi expected.…”

Classical Dynamics of Fullerenes

“…Obviously, the arrow operations satisfy the usual affine axioms [21,[47][48][49] − → ab + − → bc + − → ca = 0. (2.1) For any point p of the affine space, the operation q → − → pq is a bijection onto the translational space.…”