Euler Case Research Articles

The Chaplygin case in dynamics of a rigid body in fluid is orbitally equivalent to the Euler case in rigid body dynamics and to the Jacobi problem about geodesics on the ellipsoid

The main goal of this paper is to demonstrate how the theory of invariants for integrable Hamiltonian systems with two degrees of freedom created by A.T. Fomenko, H. Zieschang, and A.V. Bolsinov helps to establish Liouville and orbital equivalence of some classical integrable systems. Three such systems are treated in the article: the Euler case in rigid body dynamics, the Jacobi problem about geodesics on the ellipsoid and the Chaplygin case in dynamics of a rigid body in fluid. The first two systems were known to be Liouville and even topologically orbitally equivalent (Fomenko, Bolsinov). Now we show that the Chaplygin system is orbitally equivalent to the Euler and Jacobi systems.

Nonhomogeneous boundary value problem for "Equation missing" similar solutions of incompressible two-dimensional Euler equations

Abstract In this paper we introduce the ( I , J ) similar method for incompressible two-dimensional Euler equations, and obtain a series of explicit ( I , J ) similar solutions to the incompressible two-dimensional Euler equations. These solutions include all of the twin wave solutions, some new singularity solutions, and some global smooth solutions with a finite energy. We also reveal that the twin wave solution and an affine solution to the two-dimensional incompressible Euler equations are, respectively, a plane wave and constant vector. We prove that the initial boundary value problem of the incompressible two-dimensional Euler equations admits a unique solution and discuss the stability of the solution. Finally, we supply some explicit piecewise smooth solutions to the incompressible three-dimensional Euler case and an example of the incompressible three-dimensional Navier-Stokes equations which indicates that the viscosity limit of a solution to the Navier-Stokes equations does not need to be a solution to the Euler equations. MSC:35Q30, 76D05, 76D10.

A topological classification of the Chaplygin systems in the dynamics of a rigid body in a fluid

The paper is concerned with the topological analysis of the Chaplygin integrable case in the dynamics of a rigid body in a fluid. A full list of the topological types of Chaplygin systems in their dependence on the energy level is compiled on the basis of the Fomenko-Zieschang theory. An effective description of the topology of the Liouville foliation in terms of natural coordinate variables is also presented, which opens a direct way to calculating topological invariants. It turns out that on all nonsingular energy levels Chaplygin systems are Liouville equivalent to the well-known Euler case in the dynamics of a rigid body with fixed point.Bibliography: 23 titles.

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.

Contact of ropes and orthotropic rough surfaces

Contact between arbitrary curved ropes and arbitrary curved rough orthotropic surfaces has been revised from the geometrical point of view. Variational equations for the equilibrium of ropes on orthotropic rough surfaces are derived, first, using the consistent variational inclusion of frictional contact constraints via Karush‐Kuhn‐Tucker conditions expressed in Darboux basis. Then, the systems of differential equations are derived for both statics and dynamics of ropes on a rough surface depending on the sticking‐sliding condition for orthotropic Coulomb's friction. Three criteria are found to be fulfilled during the static equilibrium of a rope on a rough surface: “no separation”, condition for dragging coefficient of friction and inequality for tangential forces at the end of the rope. The limit tangential loads still preserve the famous “Euler view” T = T0e± ω s for the curves and surfaces of constant curvature. It is shown that the curve of the maximum tension of a rough orthotropic surface is geodesic. Equations of motion are derived in the case if the sliding criteria is fulfilled and there is “no separation”. Various cases possessing analytical solutions of the derived system, including Euler case and a spiral rope on a cylinder are shown as examples of application of the derived theory.

Spherically Symmetric Equilibria for Self-Gravitating Kinetic or Fluid Models in the Nonrelativistic and Relativistic Case---A Simple Proof for Finite Extension

We consider a self-gravitating collisionless gas as described by the Vlasov-Poisson or Einstein-Vlasov system or a self-gravitating fluid ball as described by the Euler-Poisson or Einstein-Euler system. We give a simple proof for the finite extension of spherically symmetric equilibria, which covers all these models simultaneously. In the Vlasov case the equilibria are characterized by a local growth condition on the microscopic equation of state, i.e., on the dependence of the particle distribution on the particle energy, at the cut-off energy E_0, and in the Euler case by the corresponding growth condition on the equation of state p=P(\rho) at \rho=0. These purely local conditions are slight generalizations to known such conditions.

Dynamics of Scaled Norms of Vorticity for the Three-dimensional Navier-Stokes and Euler Equations

A series of numerical experiments is suggested for the three-dimensional Navier-Stokes and Euler equations on a periodic domain based on a set of L2m-norms of vorticity Ωm for m ≥ 1. These are scaled to form the dimensionless sequence Dm=(ϖ0-1Ωm)αm where ϖ0 is a constant frequency and αm = 2m/(4m-3). A numerically testable Navier-Stokes regularity criterion comes from comparing the relative magnitudes of Dm and Dm+1 while another is furnished by imposing a critical lower bound on ʃ0t Dmdτ The behaviour of the Dm is also important in the Euler case in suggesting a method by which possible singular behaviour might also be tested.

Integrable cases in the dynamics of axial gyrostats and adiabatic invariants

This paper presents the study of axial gyrostats dynamics. The gyrostat is composed of two rigid bodies: an asymmetric platform and an axisymmetric rotor aligned with the platform principal axis. The paper discusses three types of gyrostats: oblate, prolate, and intermediate. Rotation of the rotor relative to the platform provides a source of small internal angular momentum and does not affect the moment of inertia tensor of the gyrostat. The dynamics of gyrostats without external torque is considered. The dynamics is described by using ordinary differential equations with Andoyer–Deprit canonical variables. For undisturbed motion, when the internal moment is equal to zero, the stationary solutions are found, and their stability is studied. General analytical solutions in terms of elliptic functions are also obtained. These results can be interpreted as the development of the classical Euler case for a solid, when to one degree of freedom—the relative rotation of bodies—is added. For disturbed motion of the gyrostats, when there is a system with slowly varying parameters, the adiabatic invariants are obtained in terms of complete elliptic integrals, which are approximately equal to the first integrals of the disturbed system. The adiabatic invariants remain approximately constant along a trajectory for long time intervals during which the parameter changes considerably. The results of the study can be useful for the analysis of dynamics of dual-spin spacecraft and for studying a chaotic behavior of the spacecraft.

External stability of resonances in the motion of an asymmetric rigid body with a strong magnet in the geomagnetic field

We consider a precession motion, close to the classical Lagrange case, of an asymmetric rigid body with a strong magnet in an orbit in the geomagnetic field. For the principal moment we take the restoring torque due to the interaction between the planet magnetic fields and the rigid body. The perturbing actions are due to small moments of the rigid body mass-inertial asymmetry and small constant moments. We show that these perturbations result in the realization of secondary resonance effects in the rotational motion of the rigid body caused by the influence of resonance denominators in higher-order approximations of the averaging method. These effects were discovered in the study of rotational motion of a satellite with a magnetic damper in the nearly Euler case. In the present paper, we analyze both the secondary resonance effects themselves and the external stability of resonances. We obtain conditions ensuring a decrease in the angular velocity of the rigid body rotation about its center of mass. We also discover several new laws of influence of resonances on the nonresonance evolution of slow variables, which is related to the appearance of stable resonances.

Stability of thin walled structures strongly coupled with contact

AbstractStability problems strongly coupled with contact are examined using different contact algorithms. Applying the penalty formulation a high dependence of the buckling load on the penalty parameter is found. This observation becomes clear when considering a classical Euler case. Applying kinematical constraint conditions (close to a Lagrange scheme) for contact on the other hand more consistent results regarding the buckling load are obtained. Thus for a Nitsche scheme a better and more flexible solution is expected. The FE implementation of a Nitsche interface element is carried out. (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Composite beam–columns with interlayer slip—Approximate analysis

An approximate second order analysis procedure for composite beam–columns with interlayer slip subjected to transverse loading and axial compressive loads is developed. The magnification factors to be applied to the first order solutions in order to estimate the deflections and internal forces obtained by the second order analysis approach are presented. The method of applying magnification factors to internal axial forces is discussed. The approximate second order analysis procedure is developed for the four Euler cases with various transverse load conditions. The procedure is applied to and the accuracy is illustrated for simply supported partially beam–columns of steel and concrete, and timber and concrete with different bending stiffness and interlayer slip properties. The deflections and internal forces obtained by the approximate method compared extremely well, except for slip forces in case of very flexible shear connectors, with those obtained by the more rigorous second order analysis approach for different composite action (partial interaction) parameters (shear connector stiffness values). The study also shows that the magnification factor associated with the deflections can be utilized to estimate also the internal actions, except shear forces in case of very flexible shear connectors, in the second order case with minimal error for simply supported beam–columns. Thus, for members with shear connector stiffness of structural significance the proposed approximate method can be used in general for simply supported beam–columns. For other boundary and loading conditions, the approximate method needs to be re-evaluated. The approach of using one magnification factor greatly simplifies the analysis task for those components.

A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes

This paper proposes a new WENO procedure to compute multi-scale problems with embedded discontinuities, on non-uniform meshes. In a one-dimensional context, the WENO procedure is first defined on a five-points stencil and designed to be fifth-order accurate in regions of smoothness. To this end, we define a finite-volume discretization in which we consider the cell averages of the variable as the discrete unknowns. The reconstruction of their point-values is then ensured by a unique fifth-order polynomial. This optimum polynomial is considered as a symmetric and convex combination, by ideal weights, of four quadratic polynomials. The symmetric nature of the resulting interpolation has an important consequence: the choice of ideal weights has no influence on the accuracy of the discretization. This advantage enables to formulate the interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, non-oscillatory weights are then calculated from the ideal weights. We adapt this procedure for the non-linear weights to maintain the theoretical convergence properties of the optimum reconstruction, whatever the problem considered. The resulting scheme is a fifth-order WENO method based on central interpolation and TVD Runge–Kutta time-integration. We call this scheme the CWENO5 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In those experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems. Finally, the new algorithm is directly extended to bi-dimensional problems.

A Hermite upwind WENO scheme for solving hyperbolic conservation laws

This paper proposes a new WENO procedure to compute problems containing both discontinuities and a large disparity of characteristic scales. In a one-dimensional context, the WENO procedure is defined on a three-points stencil and designed to be sixth-order in regions of smoothness. We define a finite-volume discretization in which we consider the cell averages of the variable and its first derivative as discrete unknowns. The reconstruction of their point-values is then ensured by a unique sixth-order Hermite polynomial. This polynomial is considered as a symmetric and convex combination, by ideal weights, of three fourth-order polynomials: a central polynomial, defined on the three-points stencil, is combined with two polynomials based on the left and the right two-points stencils. The symmetric nature of such an interpolation has an important consequence: the choice of ideal weights has no influence on the properties of the discretization. This advantage enables to formulate the Hermite interpolation for non-uniform meshes. Following the methodology of the classic WENO procedure, nonlinear weights are then defined. To deal with the peculiarities of the Hermite interpolation near discontinuities, we define a new procedure in order for the nonlinear weights to smoothly evolve between the ideal weights, in regions of smoothness, and one-sided weights, otherwise. The resulting scheme is a sixth-order WENO method based on central Hermite interpolation and TVD Runge–Kutta time-integration. We call this scheme the HCWENO6 scheme. Numerical experiments in the scalar and the 1D Euler cases make it possible to check and to validate the options selected. In these experiments, we emphasize the resolution power of the method by computing test cases that model realistic aero-acoustic problems.

Large-scale dynamics in two-dimensional Euler and surface quasigeostrophic flows

The large-scale dynamics in classical two-dimensional Euler and surface quasigeostrophic flows are studied by examining the evolution of the mean-square stream function ⟨ψ2⟩ and of the Fourier mode ψ̂(k,t) for small wave number k=∣k∣. Upper bounds for ⟨ψ2⟩ and ∣ψ̂(k,t)∣2 are derived. The growth of ⟨ψ2⟩ is at most quadratic in time t and nearly quadratic in time for surface quasigeostrophic and Euler flows, respectively. At the modal level, it is found that ∣ψ̂(k,t)∣2≤ck2t2 and ∣ψ̂(k,t)∣2≤c′t2, where c and c′ are constant, for the surface quasigeostrophic and Euler cases, respectively. These bounds imply a steep energy spectrum at small k respectively, k5 and k3. The latter is consistent with previous statistical predictions and numerical results.

Techniques for searching first integrals by Lie group and application to gyroscope system

In the paper, the methods of finding first integrals of an autonomous system using one-parameter Lie groups are discussed. A class of nontrivial one-parameter Lie groups admitted by the classical gyroscope system is found, and based on the properties of first integral determined by the one-parameter Lie group, the fourth first integral of the gyroscope system in Euler case, agrange case and Kovalevskaya case can be obtained in a uniform idea. An error on the fourth first integral in general Kovalevskaya case A = B =2 C , z G =0), which appeared in literature is found and corrected.

Euler case for a general fourth‐order differential equation

We deal with an Euler case for a general fourth‐order equation and under this case, we obtain the general formula for the asymptotic form of the solutions.

Stability analysis of hollow electron columns including compressional and thermal effects: Integrability condition and numerical simulations

The diocotron spectrum for a simplified fluid model of Malmberg–Penning traps that includes compressional effects due to end curvature with finite temperature is investigated. A class of length profiles for which the linearized eigenvalue equation for perturbations can be integrated by quadratures (integrable cases) has been found. In such cases, there is only algebraic growth when the effective angular frequency has a maximum away from the axis (hollow profile), and the model is mathematically equivalent to the zero curvature (two-dimensional Euler) case. Furthermore, profiles that are slightly nonintegrable (the difference being characterized by a small parameter ε) have been studied, finding that the complex frequency of the unstable l=1 mode scales as ε2/3. Analytical calculations (to be presented in a companion paper) and numerical simulations are found in agreement. For the density profile used, the growth rate of the unstable mode has a minimum at the plasma temperature of about 5 eV, which might be tested experimentally.

The Liouville classification of integrable systems of the Clebsch case

The Fomenko-Zieschang invariants for the integrable Clebsch case in the dynamics of a rigid body are calculated. Thus a complete description of the structure of the Liouville foliations on isoenergy surfaces of this integrable Hamiltonian system is obtained. As a corollary the Liouville equivalence of the Euler and Clebsch cases for certain values of the parameters is established.

Discussion

A dynamical system is constructed in the multiplicative group of the quarternion algebra H that serves as the configuration space. A homomorphism H → SO(3) is used such that the unit sphere S3 ⊂ H, invariant under the system, is transformed into the rotation group SO(3). The homomorphic image of the system is identical with the dynamics of rotational motion of a rigid body. The equations of motion are completely integrated in the Euler case. To this end Weierstrass' elliptic functions are used. The following goals are achieved within the framework of the method: (a) when representing the algorithms for modelling the dynamics it suffices to use only one chart from the atlas of the phase space manifold, (b) the point in the configuration space of the actual motion lies on the unit sphere, which ensures the best accuracy in numerical procedures, and (c) in the majority of applications the right-hand sides of the equations of perturbed motion depend polynomially on the phase variables, which simplifies the use of computer algebra in analytic theories.

Integration of the equations of a rotational motion of a rigid body in quaternion algebra. The Euler case

A dynamical system is constructed in the multiplicative group of the quarternion algebra H that serves as the configuration space. A homomorphism H → SO(3) is used such that the unit sphere S 3 ⊂ H, invariant under the system, is transformed into the rotation group SO(3). The homomorphic image of the system is identical with the dynamics of rotational motion of a rigid body. The equations of motion are completely integrated in the Euler case. To this end Weierstrass' elliptic functions are used. The following goals are achieved within the framework of the method: (a) when representing the algorithms for modelling the dynamics it suffices to use only one chart from the atlas of the phase space manifold, (b) the point in the configuration space of the actual motion lies on the unit sphere, which ensures the best accuracy in numerical procedures, and (c) in the majority of applications the right-hand sides of the equations of perturbed motion depend polynomially on the phase variables, which simplifies the use of computer algebra in analytic theories.