We consider two non-relativistic strings and their Galilean symmetries. These strings are obtained as the two possible non-relativistic (NR) limits of a relativistic string. One of them is non-vibrating and represents a continuum of non-relativistic massless particles, and the other one is a non-relativistic vibrating string. For both cases we write the generator of the most general point transformation and impose the condition of Noether symmetry. As a result we obtain two sets of non-relativistic Killing equations for the vector fields that generate the symmetry transformations. Solving these equations shows that NR strings exhibit two extended, infinite dimensional space-time symmetries which contain, as a subset, the Galilean symmetries. For each case, we compute the associated conserved charges and discuss the existence of non-central extensions.

The Souriau massless Galilean particle of “colour” k and spin s is shown to be the Galilean limit of the Souriau tachyon of mass m = ik and spin s. We compare and contrast this result with the Galilean limit of the Nambu-Goto string and Green-Schwarz superstring.

We construct canonical realizations of the (2+1)-dimensional Bondi-Metzner-Sachs (${\mathfrak{b}\mathfrak{m}\mathfrak{s}}_3$) algebra as symmetry algebras of a free Klein-Gordon (KG) field in $2+1$ dimensions for both massive and massless cases. We consider two types of realizations, one on shell, written in terms of the Fourier modes of the scalar field, and the other off shell, with nonlocal transformations written in terms of the KG field and its momenta. These realizations contain both supertranslations and superrotations, for which we construct the corresponding Noether charges.

We argue in this article that the standard two-stage procedure used in interconnection and damping assignment passivity-based control (IDA–PBC)–consisting of splitting the control action into the sum of energy-shaping and damping injection terms–is not without loss of generality, and effectively reduces the set of systems that can be stabilised with IDA–PBC. To overcome this problem we carry out, simultaneously, both stages and refer to this variation of the method as SIDA–PBC. To illustrate the application of SIDA–PBC we consider the practically important example given by the control problem of the induction machine. First, we show that torque and rotor flux regulation of the induction motor cannot be solved with two stage IDA–PBC. It is, however, solvable with SIDA–PBC. Second, we prove that with SIDA–PBC we can shape the total energy of the full (electrical and mechanical) dynamics of a doubly-fed induction generator used in power flow regulation tasks, while with two stage IDA–PBC only the electrical energy can be shaped. Simulation results of these examples are presented to illustrate the performance improvement obtained with SIDA–PBC.

We consider the dynamics of a particle of mass $M$ and spin $J$ in ${\mathrm{AdS}}_3$. The study reveals the presence of different dynamical sectors depending on the relative values of $M$, $J$ and the ${\mathrm{AdS}}_3$ radius $R$. For the subcritical $M^2{R}^2-J^2>0$ and supercritical $M^2{R}^2-J^2<0$ cases, it is seen that the equations of motion give the geodesics of ${\mathrm{AdS}}_3$. For the critical case $M^2{R}^2=J^2$ there exist extra gauge transformations which further reduce the physical degrees of freedom, and the motion corresponds to the geodesics of ${\mathrm{AdS}}_2$. This result should be useful in the holographic interpretation of the entanglement entropy for two-dimensional conformal field theories with gravitational anomalies.