Hamiltonian Formalism of the de-Sitter Invariant Special Relativity

Lagrangian of the Einstein's special relativity with universal parameter $c$ ($\mathcal{SR}_c$) is invariant under Poincar\'e transformation which preserves Lorentz metric $\eta_{\mu\nu}$. The $\mathcal{SR}_c$ has been extended to be one which is invariant under de Sitter transformation that preserves so called Beltrami metric $B_{\mu\nu}$. There are two universal parameters $c$ and $R$ in this Special Relativity (denote it as $\mathcal{SR}_{cR}$). The Lagrangian-Hamiltonian formulism of $\mathcal{SR}_{cR}$ is formulated in this paper. The canonic energy, canonic momenta, and 10 Noether charges corresponding to the space-time's de Sitter symmetry are derived. The canonical quantization of the mechanics for $\mathcal{SR}_{cR}$-free particle is performed. The physics related to it is discussed.Comment: 24 pages, no figur

Vortex Pinball Under Crossed AC Drives in Superconductors with Periodic Pinning Arrays

Vortices driven with both a transverse and a longitudinal AC drive which are out of phase are shown to exhibit a novel commensuration-incommensuration effect when interacting with periodic substrates. For different AC driving parameters, the motion of the vortices forms commensurate orbits with the periodicity of the pinning array. When the commensurate orbits are present, there is a finite DC critical depinning threshold, while for the incommensurate phases the vortices are delocalized and the DC depinning threshold is absent.Comment: 4 pages, 4 postscript figure

Transverse phase-locking in fully frustrated Josephson junction arrays: a new type of fractional giant steps

We study, analytically and numerically, phase locking of driven vortex lattices in fully-frustrated Josephson junction arrays at zero temperature. We consider the case when an ac current is applied {\it perpendicular} to a dc current. We observe phase locking, steps in the current-voltage characteristics, with a dependence on external ac-drive amplitude and frequency qualitatively different from the Shapiro steps, observed when the ac and dc currents are applied in parallel. Further, the critical current increases with increasing transverse ac-drive amplitude, while it decreases for longitudinal ac-drive. The critical current and the phase-locked current step width, increase quadratically with (small) amplitudes of the ac-drive. For larger amplitudes of the transverse ac-signal, we find windows where the critical current is hysteretic, and windows where phase locking is suppressed due to dynamical instabilities. We characterize the dynamical states around the phase-locking interference condition in the $IV$ curve with voltage noise, Lyapunov exponents and Poincar\'e sections. We find that zero temperature phase-locking behavior in large fully frustrated arrays is well described by an effective four plaquette model.Comment: 12 pages, 11 figure

Transverse Phase Locking for Vortex Motion in Square and Triangular Pinning Arrays

We analyze transverse phase locking for vortex motion in a superconductor with a longitudinal DC drive and a transverse AC drive. For both square and triangular arrays we observe a variety of fractional phase locking steps in the velocity versus DC drive which correspond to stable vortex orbits. The locking steps are more pronounced for the triangular arrays which is due to the fact that the vortex motion has a periodic transverse velocity component even for zero transverse AC drive. All the steps increase monotonically in width with AC amplitude. We confirm that the width of some fractional steps in the square arrays scales as the square of the AC driving amplitude. In addition we demonstrate scaling in the velocity versus applied DC driving curves at depinning and on the main step, similar to that seen for phase locking in charge-density wave systems. The phase locking steps are most prominent for commensurate vortex fillings where the interstitial vortices form symmetrical ground states. For increasing temperature, the fractional steps are washed out very quickly, while the main step gains a linear component and disappears at melting. For triangular pinning arrays we again observe transverse phase locking, with the main and several of the fractional step widths scaling linearly with AC amplitude.Comment: 10 pages, 14 postscript figure