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A rotary motion is studied, which is used as a paradigm in both mathematics and dynamics. In this motion, the body moves around its axis. Important examples of this motion are the earth, wheels for cars or bicycles, cooling fans, airplanes, radars, and motion of the discs. In this article, we consider one of these motions such as the slow rotational motion of a disc of high natural frequency around a fixed point different from its center of mass. Suppose a constant gyrostatic couple acts on the body about the axis of symmetry. Let us consider the axis of symmetry coincide with one of the main axes of inertia. The controlled nonlinear independent system of equations of motion is obtained in the form of six nonlinear differential equations in the presence of three independent first integrals. This system is reduced to an alternative independent quasi-linear system consisting of two differential equations with only one integral. At first, when the body rotates slowly with a small angular speed about the axis of the symmetry, we achieve a large parameter for the motion which is proportional to the inverse of the angular velocity component. Then, we use the large parameter technique to obtain approximate and high-frequency analytic solutions for this motion. This technique enables us to introduce new conditions of the motion, save a lot of energy required to move the disc, and obtain new approximated solutions in a new domain of the disc parameters. We present a digital example of the problem to clarify the inferences of the motion depending on the parameters of the body. On the other hand, we use the Runge–Kutta algorithmic method to obtain the numerical solutions corresponding to the approximate solutions of the considered independent system. We investigate the inaccuracies and errors of analytic and numerical solutions by comparing them. This problem offers many applications in gyroscope dynamics, industrial engineering, astrophysics, navigation, satellites, and space engineering. [ABSTRACT FROM AUTHOR]

We study axially symmetric Euler's system of equations for the real gas in two space dimensions. This problem poses various difficulties for arbitrary equation of state and general problem is still open. Deviation from ideal gas law generally involves change of phase and more number of chemical species. This affects number of degrees of freedom which is governed by phase rule. We analyze self-similar solutions of Euler's system of equations for the real gas (alternatively called non-ideal gas) which is in one phase but may contain vapor or moisture. We find that structure of solutions in this case is governed by fundamental derivative of underlying gas. Emergence of fundamental derivative in calculations carried out by the authors suggest that structure of solutions to Euler's equations in gas dy namics is fundamentally dependent on the thermodynamics of underlying gas. We have considered case of zero swirl in this article. [ABSTRACT FROM AUTHOR]

The purpose of this study is essentially pedagogical and aims to provide an additional argument in clarification of a question often raised by first-year undergraduate mechanical engineering students concerning the reason for using two frames of reference—one fixed in space and one fixed in the rigid-body—to describe its motion. The reasoning employed to illustrate the inappropriateness of using a single reference frame entails showing that the equations of motion, thus obtained, are far more complex than the equations resulting from application of the traditional Euler Method. This point is illustrated through the well-known frictionless symmetrical spinning top problem. [ABSTRACT FROM AUTHOR]

The motion of a rigid body is described in Classical Mechanics with the venerable Euler's equations which are based on the assumption that the relative distances among the constituent particles are fixed in time. Real bodies, however, cannot satisfy this property, as a consequence of thermal fluctuations. We generalize Euler's equations for a free body in order to describe dissipative and thermal fluctuation effects in a thermodynamically consistent way. The origin of these effects is internal, i.e. not due to an external thermal bath. The stochastic differential equations governing the orientation and central moments of the body are derived from first principles through the theory of coarse-graining. Within this theory, Euler's equations emerge as the reversible part of the dynamics. For the irreversible part, we identify two distinct dissipative mechanisms; one associated with diffusion of the orientation, whose origin lies in the difference between the spin velocity and the angular velocity, and one associated with the damping of dilations, i.e. inelasticity. We show that a deformable body with zero angular momentum will explore uniformly, through thermal fluctuations, all possible orientations. When the body spins, the equations describe the evolution towards the alignment of the body's major principal axis with the angular momentum vector. In this alignment process, the body increases its temperature. We demonstrate that the origin of the alignment process is not inelasticity but rather orientational diffusion. The theory also predicts the equilibrium shape of a spinning body. • Euler's equations are generalized to include dissipation and thermal fluctuations. • Thermal fluctuations induce change in orientation of small non-spinning bodies. • Nutation damping leading to re-orientation of the body to its major axis is captured by the theory. • Nutation damping results from orientational diffusion, not body viscoelasticity. [ABSTRACT FROM AUTHOR]

This work introduces a hybridizable discontinuous Galerkin formulation for the simulation of ideal plasmas governed by the Euler–Maxwell equations. The approach is based on a monolithic source-based coupling of the fluid and electromagnetic subproblems, along with a fully implicit time integration scheme, and a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. The numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation and validate it against problems of increasing complexity, ranging from single-physics cases to weakly and fully coupled electromagnetic plasma simulations. • Novel Hybridizable Discontinuous Galerkin formulation for Euler-Maxwell plasma model. • Monolithic source-based coupling and fully implicit time integration scheme. • Projection-based divergence correction method to enforce Gauss' laws. • Simulation of high-frequency plasma phenomena. • Accuracy, efficiency and robustness demonstrated on problems of increasing complexity. [ABSTRACT FROM AUTHOR]

The recently proposed low degree-of-freedom model of Moffatt and Kimura [1,2] for describing the approach to finite-time singularity of the incompressible Euler fluid equations is investigated. The model assumes an initial finite-energy configuration of two vortex rings placed symmetrically on two tilted planes. The Hamiltonian structure of the inviscid limit of the model is obtained. The associated noncanonical Poisson bracket [3] and two invariants, one that serves as the Hamiltonian and the other a Casimir invariant, are discovered. It is shown that the system is integrable with a solution that lies on the intersection for the two invariants, just as for the free rigid body of mechanics whose solution lies on the intersection of the kinetic energy and angular momentum surfaces. Also, a direct quadrature is given and used to demonstrate the Leray form for finite-time singularity in the model. To the extent the Moffatt and Kimura model accurately represents Euler's ideal fluid equations of motion, we have shown the existence of finite-time singularity. • A low degree-of freedom model for the interaction of two tilted vortex rings is described. • Noncanonical Hamiltonian structure is described, yielding two previously unknown invariants, the Hamiltonian and Casimir. • The Hamiltonian system is integrable providing analytical and numerical solutions for all initial conditions. • Leray-like solutions that diverge in finite time are given, suggesting finite-time singularity for Euler's fluid equations. [ABSTRACT FROM AUTHOR]