An extension to the theory of controlled Lagrangians using the Helmholtz conditions

The Helmholtz conditions are necessary and sufficient conditions for a system of second order differential equations to be variational, that is, equivalent to a system of Euler-Lagrange equations for a regular Lagrangian. On the other hand, matching conditions are sufficient conditions for a class of controlled systems to be variational for a Lagrangian function of a prescribed type, known as the controlled Lagrangian. Using the Helmholtz conditions we are able to recover the matching conditions from [8]. Furthermore we can derive new matching conditions for a particular class of mechanical systems. It turns out that for this class of systems we obtain feedback controls that only depend on the configuration variables. We test this new strategy for the inverted pendulum on a cart and for the inverted pendulum on an incline

Evolution of Planetary Orbits with Stellar Mass Loss and Tidal Dissipation

Intermediate mass stars and stellar remnants often host planets, and these dynamical systems evolve because of mass loss and tides. This paper considers the combined action of stellar mass loss and tidal dissipation on planetary orbits in order to determine the conditions required for planetary survival. Stellar mass loss is included using a so-called Jeans model, described by a dimensionless mass loss rate \gamma and an index \beta. We use an analogous prescription to model tidal effects, described here by a dimensionless dissipation rate \Gamma and two indices (q,p). The initial conditions are determined by the starting value of angular momentum parameter \eta (equivalently, the initial eccentricity) and the phase \theta of the orbit. Within the context of this model, we derive an analytic formula for the critical dissipation rate \Gamma, which marks the boundary between orbits that spiral outward due to stellar mass loss and those that spiral inward due to tidal dissipation. This analytic result \Gamma=\Gamma(\gamma,\beta,q,p,\eta,\theta) is essentially exact for initially circular orbits and holds to within an accuracy of 50% over the entire multi-dimensional parameter space, where the individual parameters vary by several orders of magnitude. For stars that experience mass loss, the stellar radius often displays quasi-periodic variations, which produce corresponding variations in tidal forcing; we generalize the calculation to include such pulsations using a semi-analytic treatment that holds to the same accuracy as the non-pulsating case. These results can be used in many applications, e.g., to predict/constrain properties of planetary systems orbiting white dwarfs.Comment: 17 pages, 4 figures, accepted to ApJ Letter

On the Stability of Extrasolar Planetary Systems and other Closely Orbiting Pairs

This paper considers the stability of tidal equilibria for planetary systems in which stellar rotation provides a significant contribution to the angular momentum budget. We begin by applying classic stability considerations for two bodies to planetary systems --- where one mass is much smaller than the other. The application of these stability criteria to a subset of the Kepler sample indicates that the majority of the systems are not in a stable equilibrium state. Motivated by this finding, we generalize the stability calculation to include the quadrupole moment for the host star. In general, a stable equilibrium requires that the total system angular momentum exceeds a minimum value (denoted here as $L_X$) and that the orbital angular momentum of the planet exceeds a minimum fraction of the total. Most, but not all, of the observed planetary systems in the sample have enough total angular momentum to allow an equilibrium state. Even with the generalizations of this paper, however, most systems have too little orbital angular momentum (relative to the total) and are not in an equilibrium configuration. Finally, we consider the time evolution of these planetary systems; the results constrain the tidal quality factor of the stars and suggest that $10^6\le{Q_\ast}\le10^7$.Comment: 13 pages, 9 figures, accepted to MNRA

Optimal Control of Underactuated Nonholonomic Mechanical Systems

In this paper we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, under-actuated mechanical systems. In particular, we aim at minimizing the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and under-actuation. Nonholonomic constraints restrict the evolution of the system to a distribution on the manifold. The nonholonomic connection is used to express the constrained equations of motion. Furthermore, it is used to take variations of the cost functional. Many robotic systems are under-actuated since control inputs are usually applied through the robot's internal configuration space only. While we do not consider symmetries with respect to group actions in this paper, the fact that the system is under-actuated is taken into account in our problem formulation. This allows one to compute reaction forces due to any inputs applied in directions orthogonal to the constraint distribution. We illustrate our ideas by considering a simple example on a three-dimensional manifold.Comment: 8 pages, 1 figur

The rolling sphere and the quantum spin

We consider the problem of a sphere rolling of a curved surface and solve it by mapping it to the precession of a spin 1/2 in a magnetic field of variable magnitude and direction. The mapping can be of pedagogical use in discussing both rolling and spin precession, and in particular in understanding the emergence of geometrical phases in classical problems

Hill's Equation with Small Fluctuations: Cycle to Cycle Variations and Stochastic Processes

Hill's equations arise in a wide variety of physical problems, and are specified by a natural frequency, a periodic forcing function, and a forcing strength parameter. This classic problem is generalized here in two ways: [A] to Random Hill's equations which allow the forcing strength q_k, the oscillation frequency \lambda_k, and the period \tau_k of the forcing function to vary from cycle to cycle, and [B] to Stochastic Hill's equations which contain (at least) one additional term that is a stochastic process \xi. This paper considers both random and stochastic Hill's equations with small parameter variations, so that p_k=q_k-, \ell_k=\lambda_k-, and \xi are all O(\epsilon), where \epsilon<<1. We show that random Hill's equations and stochastic Hill's equations have the same growth rates when the parameter variations p_k and \ell_k obey certain constraints given in terms of the moments of \xi. For random Hill's equations, the growth rates for the solutions are given by the growth rates of a matrix transformation, under matrix multiplication, where the matrix elements vary from cycle to cycle. Unlike classic Hill's equations where the parameter space (the \lambda-q plane) displays bands of stable solutions interlaced with bands of unstable solutions, random Hill's equations are generically unstable. We find analytic approximations for the growth rates of the instability; for the regime where Hill's equation is classically stable, and the parameter variations are small, the growth rate \gamma = O(\epsilon^2). Using the relationship between the (\ell_k,p_k) and the \xi, this result for \gamma can be used to find growth rates for stochastic Hill's equations.Comment: 22 pages, 3 figures, accepted to Journal of Mathematical Physic

Flag-Based Control of Quantum Purity for n=2n=2 Systems

This paper investigates the fast Hamiltonian control of $n=2$ density operators by continuously varying the flag as one moves away from the completely mixed state. In general, the critical points and zeros of the purity derivative can only be solved analytically in the limit of minimal purity. We derive differential equations that maintain these features as the purity increases. In particular, there is a thread of points in the Bloch ball that locally maximizes the purity derivative, and a corresponding thread that minimizes it. Additionally, we show there is a closed surface of points inside of which the purity derivative is positive, and inside of which is negative. We argue that this approach may be useful in studying higher-dimensional systems.Comment: 8 pages, 4 figure

Flag-based Control of Orbit Dynamics in Quantum Lindblad Systems

In this paper, we demonstrate that the dynamics of an $n$-dimensional Lindblad control system can be separated into its inter- and intra-orbit dynamics when there is fast controllability. This can be viewed as a control system on the simplex of density operator spectra, where the flag representing the eigenspaces is viewed as a control variable. The local controllability properties of this control system can be analyzed when the control-set of flags is limited to a finite subset. In particular, there is a natural finite subset of $n!$ flags that are effective for low-purity orbits.Comment: 13 pages, 5 figure

Trees, Forests, and Stationary States of Quantum Lindblad Systems

In this paper, we study the stationary orbits of quantum Lindblad systems. We show that they can be characterized in terms of trees and forests on a directed graph with edge weights that depend on the Lindblad operators and the eigenbasis of the density operator. For a certain class of typical Lindblad systems, this characterization can be used to find the asymptotic end-states. There is a unique end-state for each basin of the graph (the strongly connected components with no outgoing edges). In most cases, every asymptotic end-state must be a linear combination thereof, but we prove necessary and sufficient conditions under which symmetry in the Lindblad and Hamiltonian operators hide other end-states or stable oscillations between end-states.Comment: 17 pages, 7 figure

Turbulence in Extrasolar Planetary Systems Implies that Mean Motion Resonances are Rare

This paper considers the effects of turbulence on mean motion resonances in extrasolar planetary systems and predicts that systems rarely survive in a resonant configuration. A growing number of systems are reported to be in resonance, which is thought to arise from the planet migration process. If planets are brought together and moved inward through torques produced by circumstellar disks, then disk turbulence can act to prevent planets from staying in a resonant configuration. This paper studies this process through numerical simulations and via analytic model equations, where both approaches include stochastic forcing terms due to turbulence. We explore how the amplitude and forcing time intervals of the turbulence affect the maintenance of mean motion resonances. If turbulence is common in circumstellar disks during the epoch of planet migration, with the amplitudes indicated by current MHD simulations, then planetary systems that remain deep in mean motion resonance are predicted to be rare. More specifically, the fraction of resonant systems that survive over a typical disk lifetime of 1 Myr is of order 0.01. If mean motion resonances are found to be common, their existence would place tight constraints on the amplitude and duty cycle of turbulent fluctuations in circumstellar disks. These results can be combined by expressing the expected fraction of surviving resonant systems in the approximate form P_b = C / N_{orb}^{1/2}, where the dimensionless parameter C = 10 - 50 and where N_{orb} is the number of orbits for which turbulence is active.Comment: 30 pages, 5 figures, accepted to Ap