We introduce a formalism for the calculation of the time of arrival t at a space point for particles traveling through interacting media. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t in the context of the positive-operator-valued measures. We then compute the probability distribution in the times of arrival at a point for particles that have undergone reflection, transmission or tunneling off finite potential barriers. For narrow Gaussian initial wave packets we obtain multimodal time distribution of the reflected packets and a combination of the Hartman effect with unexpected retardation in tunneling. We also employ explicitly our formalism to deal with arrivals in the interaction region for the step and linear potentials.

In this note we revisit E. Cartan's address at the 1928 International Congress of Mathematicians at Bologna, Italy. The distributions considered here will be of the same class as those considered by Cartan, a special type which we call strongly or maximally non-holonomic. We set up the groundwork for using Cartan's method of equivalence (a powerful tool for obtaining invariants associated to geometrical objects), to more general non-holonomic distributions.

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%'e study the theory for the structure of coupled incommensurate chains in the weak-coupling limit. Both chains are deformable. The emphasis is on the commensurate-incommensurate transition. In this limit, the structure is described in terms of commensurate domains separated by walls. which have intrachainand interchain-interactions. It is argued that the latter cause defects of one chain to be pinned on defects of the other, so that the commensurate-incommensurate transition is in general complicated and exhibits a behavior reminiscent of the devil' s staircase. %e show that crystals formed from interpenetrating chains may have different types of commensurate-incommensurate transitions, depending on whether one sublattice only forms defects (walls) at the transition, or both simultaneously.In the last section we discuss the fluctuations of defect planes at low temperatures in quasi-one-dimensional conductors.on the simple idea that in the commensurateincommensurate limit, everything can be described in terms of commensurate domains separated by walls, which have intrachain interactions and interchain in-23

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One can develop a symplectic reduction procedure for higher order Lagrangian systems with symmetry. The reconstruction procedure of the dynamics is also studied and an application (spinning particle) is given at the end of the work.