Casimir Functions Research Articles

Equilibrium attitude and nonlinear attitude stability of a spacecraft on a stationary orbit around an asteroid

The classical problem of attitude stability in a central gravity field is generalized to that on a stationary orbit around a uniformly-rotating asteroid. This generalized problem is studied in the framework of geometric mechanics. Based on the natural symplectic structure, the non-canonical Hamiltonian structure of the problem is derived. The Poisson tensor, Casimir functions and equations of motion, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. The equilibrium of the equations of motion, i.e. the equilibrium attitude of the spacecraft, which corresponds to a stationary point of the Hamiltonian constrained by Casimir functions, is determined from a global point of view. Nonlinear stability conditions of the equilibrium attitude are obtained in a modified energy-Casimir method. The nonlinear attitude stability is then investigated versus three parameters of the asteroid, including the ratio of the mean radius to the orbital radius, the harmonic coefficients C20 and C22. We find that when the spacecraft is located on the intermediate-moment principal axis of the asteroid, the nonlinear stability domain can be totally different from the classical Lagrange region on a circular orbit in a central gravity field. With the Lie group framework uncovered by geometric mechanics, besides determination of the equilibrium from a global point of view and the energy-Casimir method for nonlinear stability, several other powerful techniques can be performed, such as the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

An energy-preserving description of nonlinear beam vibrations in modal coordinates

Conserved quantities are identified in the equations describing large-amplitude free vibrations of beams projected onto their linear normal modes. This is achieved by writing the geometrically exact equations of motion in their intrinsic, or Hamiltonian, form before the modal transformation. For nonlinear free vibrations about a zero-force equilibrium, it is shown that the finite-dimensional equations of motion in modal coordinates are energy preserving, even though they only approximate the total energy of the infinite-dimensional system. For beams with constant follower forces, energy-like conserved quantities are also obtained in the finite-dimensional equations of motion via Casimir functions. The duality between space and time variables in the intrinsic description is finally carried over to the definition of a conserved quantity in space, which is identified as the local cross-sectional power. Numerical examples are used to illustrate the main results.

On Casimir Functionals for Infinite-Dimensional Port-Hamiltonian Control Systems

We consider infinite-dimensional port-Hamiltonian systems with respect to control issues. In contrast to the well-established representation relying on Stokes-Dirac structures that are based on skew-adjoint differential operators and the use of energy variables, we employ a different port-Hamiltonian framework. Based on this system representation conditions for Casimir functionals will be derived where in this context the variational derivative plays an extraordinary role. Furthermore the coupling of finite- and infinite-dimensional systems will be analyzed in the spirit of the control by interconnection problem.

Stability of the classical type of relative equilibria of a rigid body in the J 2 problem

The motion of a point mass in the J 2 problem is generalized to that of a rigid body in a J 2 gravity field. The linear and nonlinear stability of the classical type of relative equilibria of the rigid body, which have been obtained in our previous paper, are studied in the framework of geometric mechanics with the second-order gravitational potential. Non-canonical Hamiltonian structure of the problem, i.e., Poisson tensor, Casimir functions and equations of motion, are obtained through a Poisson reduction process by means of the symmetry of the problem. The linear system matrix at the relative equilibria is given through the multiplication of the Poisson tensor and Hessian matrix of the variational Lagrangian. Based on the characteristic equation of the linear system matrix, the conditions of linear stability of the relative equilibria are obtained. The conditions of nonlinear stability of the relative equilibria are derived with the energy-Casimir method through the projected Hessian matrix of the variational Lagrangian. With the stability conditions obtained, both the linear and nonlinear stability of the relative equilibria are investigated in details in a wide range of the parameters of the gravity field and the rigid body. We find that both the zonal harmonic J 2 and the characteristic dimension of the rigid body have significant effects on the linear and nonlinear stability. Similar to the classical attitude stability in a central gravity field, the linear stability region is also consisted of two regions that are analogues of the Lagrange region and the DeBra-Delp region respectively. The nonlinear stability region is the subset of the linear stability region in the first quadrant that is the analogue of the Lagrange region. Our results are very useful for the studies on the motion of natural satellites in our solar system.

On the relationship between the classical Dicke-Jaynes-Cummings-Gaudin model and the nonlinear Schrödinger equation

In this paper, the relationship between the classical Dicke-Jaynes-Cummings-Gaudin (DJCG) model and the nonlinear Schrödinger (NLS) equation is studied. It is shown that the classical DJCG model is equivalent to a stationary NLS equation. Moreover, the standard NLS equation can be solved by the classical DJCG model and a suitably chosen higher order flow. Further, it is also shown that classical DJCG model can be transformed into the classical Gaudin spin model in an external magnetic field through a deformation of Lax matrix. Finally, the separated variables are constructed on the common level sets of Casimir functions and the generalized action-angle coordinates are introduced via the Hamilton-Jacobi equation.

Casimir method for attitude stability analysis of liquid-filled spacecraft

Sloshing is the phenomenon which is related with the free surface movement of fluid inside a container. In this paper, Hamiltonian dynamics method was used to study the stability of attitude motion for spacecraft with liquid sloshing in tank .In this article, the equivalent mass-spring mechanical model was used to analogue the liquid sloshing dynamics which was coupled with spacecraft attitude dynamics. Hamiltonian equations and Casimir functions was found for the proposed dynamical model for the coupled multi-body liquid-filled spacecraft system. The stability conditions and instability conditions for attitude motion of partially liquid-filled spacecraft were derived by using Lyapunov function along with Casimir functions. Numerical result was presented to strengthen the derived results and to show the geometric illustration of the stable and unstable regions in parameters space.

A complete list of conservation laws for non-integrable compacton equations of K(m, m) type

In 1993, P Rosenau and J M Hyman introduced and studied Korteweg–de-Vries-like equations with nonlinear dispersion admitting compacton solutions, , m, n > 1, which are known as the K(m, n) equations. In this paper we consider a slightly generalized version of the K(m, n) equations for m = n, namely, , where m, a, b are arbitrary real numbers. We describe all generalized symmetries and conservation laws thereof for m ≠ −2, −1/2, 0, 1; for these four exceptional values of m the equation in question is either completely integrable (m = −2, −1/2) or linear (m = 0, 1). It turns out that for m ≠ −2, −1/2, 0, 1 there are only three symmetries corresponding to x- and t-translations and scaling of t and u, and four non-trivial conservation laws, one of which expresses the conservation of energy, and the other three are associated with the Casimir functionals of the Hamiltonian operator admitted by our equation. Our result provides inter alia a rigorous proof of the fact that the K (2, 2) equation has just four conservation laws from the paper of P Rosenau and J M Hyman.

Dirac Structures on Hilbert Spaces and Boundary Control of Distributed Port-Hamiltonian Systems

The scope of this paper is to show how to exploit the properties of the Stokes–Dirac structure in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Usually, stabilisation of non-zero equilibria has been achieved by shaping the open-loop Hamiltonian function by interconnecting a finite-dimensional port-Hamiltonian controller to the boundary of the distributed parameter system. The procedure is based on the presence of structural invariants, namely Casimir functions, in closed-loop, but this approach fails when a non-zero power flow from the controller is required at the equilibrium (dissipation obstacle). This paper illustrates that the class of stabilising controllers can be enlarged by relying on the parametrisation of the system dynamics provided by the image representation of the Stokes–Dirac structure, that is able to show the effects of the boundary inputs on the state evolution in a simple and effective way.

Hamiltonian Structure and Stability Analysis for a Partially Filled Container

AbstractHamiltonian system is a special case of dynamical system. Mostly it is used for potential shaping of mechanical systems stabilization. In our present work, we are using Hamiltonian dynamics to study and control the fuel slosh inside spacecraft tank. Sloshing is the phenomenon which is related with the movement of fluid inside a container in micro and macro scale as well. Sloshing of fluid occurs whenever the frequency of container movement matches with the natural frequency of fluid inside the container. Such type of synchronization may cause the structural damage or could be a reason of moving object's attitude disturbance. In spacecraft technology, the equivalent mechanical model for sloshing is common to use for the representation of fuel slosh. This mechanical model may contain a model of pendulum or a mass attached with a spring. In this article, we are using mass-spring mechanical model coupled with rigid body to derive the equations for Hamiltonian system. Casimir functions are used for proposed model. Conditions for the stability and instability of moving mass are derived using Lyapunov function along with Casimir functions. Simulation work is presented to strengthen the derived results and to distribute the stable and unstable regions graphically.

Poisson Brackets with Prescribed Casimirs

Abstract We consider the problem of constructing Poisson brackets on smooth manifolds M with prescribed Casimir functions. If M is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on M, while, in the case where M is of odd dimension, our objective is achieved using a convenient almost cosymplectic structure. Several examples and applications are presented.

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.

Classification of real three-dimensional Poisson–Lie groups

All real three-dimensional Poisson–Lie (PL) groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in Gomez (2000 J. Math. Phys. 41 4939). Many of these 3D PL groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented.

On the Recurrence and Robust Properties of Lorenz’63 Model

Lie-Poisson structure of the Lorenz'63 system gives a physical insight on its dynamical and statistical behavior considering the evolution of the associated Casimir functions. We study the invariant density and other recurrence features of a Markov expanding Lorenz-like map of the interval arising in the analysis of the predictability of the extreme values reached by particular physical observables evolving in time under the Lorenz'63 dynamics with the classical set of parameters. Moreover, we prove the statistical stability of such an invariant measure. This will allow us to further characterize the SRB measure of the system.

Fast computation by simplifications of a class of hydro-mechanical systems

This paper presents a fast computation method of forward dynamics of a class of hydro-mechanical systems. First, an exact simplification of the conventional representation of the hydro-mechanical systems is presented based on Casimir functions. Second, a further simplification is given via an integration by parts which is a new structural property and a new representation of the hydro-mechanical systems is proposed. Third, the proposed representation and the conventional representation are compared with each other with respect to computational cost and the validity of the proposed representation is confirmed by numerical experiment.

Boundary Energy Shaping of Linear Distributed Port-Hamiltonian Systems

This paper deals with the stabilization via Casimir generation and energy shaping of linear, lossless, distributed port-Hamiltonian systems. Once inputs and outputs of the distributed port-Hamiltonian system have been chosen to obtain a well-defined boundary control systems, conditions for the existence of Casimir functions in closed-loop and of the associated semigroup are given, together with a criterion to be used to check asymptotic stability. Casimir functions suggest how to select the controller Hamiltonian to introduce a minimum at the desired equilibrium, while stability is ensured if proper “pervasive” boundary damping is present. The methodology is illustrated with the help of a Timoshenko beam with full-actuation on one side.

Casimir-Based Control Beyond the Dissipation Obstacle

A prevailing trend in the stabilization of port-Hamiltonian systems is the assumption that the plant and the controller are both passive. In the standard approach of control by interconnection based on the generation of Casimir functions, this assumption leads to the dissipation obstacle, which essentially means that dissipation is admissible only on the coordinates of the closed-loop Hamiltonian that do not require shaping and thus severely restricts the scope of applications. In this contribution, we show that we can easily go beyond the dissipation obstacle by allowing the controller to have a negative semi-definite resistive structure, while guaranteeing stability of both the closed-loop and the controller.

Hamilton–Poisson formulation for the rotational motion of a rigid body in the presence of an axisymmetric force field and a gyroscopic torque

We give sufficient conditions for the rigid body in the presence of an axisymmetric force field and a gyroscopic torque to admit a Hamilton–Poisson formulation. Even if by adding a gyroscopic torque we initially lose one of the conserved Casimirs, we recover another conservation law as a Casimir function for a modified Poisson structure. We apply this frame to several well-known results in the literature.

Development of level set method with good area preservation to predict interface in two‐phase flows

AbstractA two‐step conservative level set method is proposed in this study to simulate the gas/water two‐phase flow. For the sake of accuracy, the spatial derivative terms in the equations of motion for an incompressible fluid flow are approximated by the coupled compact scheme. For accurately predicting the modified level set function, the dispersion‐relation‐preserving advection scheme is developed to preserve the theoretical dispersion relation for the first‐order derivative terms shown in the pure advection equation cast in conservative form. For the purpose of retaining its long‐time accurate Casimir functionals and Hamiltonian in the transport equation for the level set function, the time derivative term is discretized by the sixth‐order accurate symplectic Runge–Kutta scheme. To resolve contact discontinuity oscillations near interface, nonlinear compression flux term and artificial damping term are properly added to the second‐step equation of the modified level set method. For the verification of the proposed dispersion‐relation‐preserving scheme applied in non‐staggered grids for solving the incompressible flow equations, three benchmark problems have been chosen in this study. The conservative level set method with area‐preserving property proposed for capturing the interface in incompressible fluid flows is also verified by solving the dam‐break, Rayleigh–Taylor instability, bubble rising in water, and droplet falling in water problems. Good agreements with the referenced solutions are demonstrated in all the investigated problems. Copyright © 2010 John Wiley & Sons, Ltd.

Energy shaping of distributed parameter port-Hamiltonian systems based on finite element approximation

The main contribution of this paper is a procedure for the control by energy shaping via Casimir generation of infinite dimensional port-Hamiltonian systems based on a particular finite element approximation. The proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve Casimir functions in the closed-loop system when going from the distributed to the (approximated) lumped parameter system. Besides the intrinsic difficulties related to the large number of state variables, the finite element model is generally given in terms of a Dirac structure and is completely a - causal, which implies that the plant dynamics is not given in standard input-state-output form, but as a set of DAEs. Consequently, the classical energy Casimir method has to be extended in order to deal with dynamical systems with constraints, usually appearing in the form of Lagrangian multipliers. The general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line.

Linear energy-preserving integrators for Poisson systems

For Hamiltonian systems with non-canonical structure matrix a new class of numerical integrators is proposed. The methods exactly preserve energy, are invariant with respect to linear transformations, and have arbitrarily high order. Those of optimal order also preserve quadratic Casimir functions. The discussion of the order is based on an interpretation as partitioned Runge-Kutta method with infinitely many stages.