We consider two types of trajectories found in a wide range of mechanical systems, viz. box orbits and loop orbits. We elucidate the dynamics of these orbits in the simple context of a perturbed harmonic oscillator in two dimensions. We then examine the small-amplitude motion of a rigid body, the rock’n’roller, a sphere with eccentric distribution of mass. The equations of motion are expressed ...

Phase mixing of chaotic orbits exponentially distributes these orbits through their accessible phase space. This phenomenon, commonly called ‘‘chaotic mixing,’’ stands in marked contrast to phase mixing of regular orbits which proceeds as a power law in time. It is operationally irreversible; hence, its associated e-folding time scale sets a condition on any process envisioned for emittance com...

Orbits of the coadjoint action play an important role in harmonic analysis, theory of dynamic systems, noncommutative geometry. A.A. Kirillov’s orbit method allows to reduce the classification problem of unitary representations of nilpotent Lie groups to the classification of coadjoint orbits [1, 2]. This makes possible to solve the problems of representation theory in geometric terms of the or...

Let T be an operator on a Banach space X. We give a survey of results concerning orbits {Tnx : n = 0, 1, . . .} and weak orbits {〈Tnx, x∗〉 : n = 0, 1, . . .} of T where x ∈ X and x∗ ∈ X∗. Further we study the local capacity of operators and prove that there is a residual set of points x ∈ X with the property that the local capacity cap(T, x) is equal to the global capacity cap T . This is an an...

The photoabsorption spectra of highly excited Rydberg atoms in external fields are known to be related to closed classical orbits of the excited electron. For atoms in a pure magnetic field, scaled recurrence spectra have successfully been calculated from classical orbits using the semiclassical closed-orbit theory. Rydberg atoms in crossed fields are more complicated as regards their theoretic...

We carry out a set of self-consistent N -body calculations to investigate how important the velocity anisotropy in non-spherical dark-matter halos is for dynamical friction. For this purpose we allow satellite galaxies to orbit within flattened and live dark-matter haloes (DMHs) and compare the resulting orbit evolution with a semi-analytic code. This code solves the equation of motion of the s...

A technique for real-time, wide-area differential kinematic positioning at the decimeter level has been tested with data from GPS stations in Europe and the USA, streamed in real time over the Internet by various organizations using the protocol known as NTRIP (Network Transport of RTCM via Internet Protocol). With the technique presented here, implemented in navigation software developed by th...

Maslov indices in periodic-orbit theory are investigated using phase space path integral. Based on the observation that the Maslov index is the multi-valued function of the monodromy matrix, we introduce a generalized monodromy matrix in the universal covering space of the symplectic group and show that this index is uniquely determined in this space. The stability of the orbit is shown to dete...

By using the foundational solutions of the linear variational equation of the unperturbed system along the heteroclinic orbits to establish the local coordinate systems in the small tubular neighborhoods of the heteroclinic orbits, we study the bifurcation problems of nontwisted heteroclinic loop with resonant eigenvalues. The existence, numbers, and existence regions of 1-heteroclinic loop, 1-...

Based on symbolic dynamics, the paper provides a satisfactory and necessary condition of existence for consecutive periodic orbits of the Lorenz maps. In addition, a new algorithm with computer assistance based on symbolic dynamics is proposed to find all periodic orbits up to a certain number with little computer time. Examples for consecutive periods of orbits are raised for the Lorenz maps. ...