Showing papers on "Equations of motion published in 1979"

A mathematical introduction to fluid mechanics

Abstract: Contents: The Equations of Motion.- Potential Flow and Slightly Viscous Flow.- Gas Flow in One Dimension.- Vector Identities.- Index.

A classical analog for electronic degrees of freedom in nonadiabatic collision processes

Abstract: It is shown how a formally exact classical analog can be defined for a finite dimensional (in Hilbert space) quantum mechanical system. This approach is then used to obtain a classical model for the electronic degrees of freedom in a molecular collision system, and the combination of this with the usual classical description of the heavy particle (i.e., nuclear) motion provides a completely classical model for the electronic and heavy particle degrees of freedom. The resulting equations of motion are shown to be equivalent to describing the electronic degrees of freedom by the time‐dependent Schrodinger equation, the time dependence arising from the classical motion of the nuclei, the trajectory of which is determined by the quantum mechanical average (i.e., Ehrenfest) force on the nuclei. Quantizing the system via classical S‐matrix theory is shown to provide a dynamically consistent description of nonadiabatic collision processes; i.e., different electronic transitions have different heavy particle trajectories and, for example, the total energy of the electronic and heavy particle degrees of freedom is conserved. Application of this classical model for the electronic degrees of freedom (plus classical S‐matrix theory) to the two‐state model problem shows that the approach provides a good description of the electronic dynamics.

The Initial Value Problem for the Equations of Motion of compressible Viscous and Heat-conductive Fluids.

Abstract: : The initial value problem associated with the equations of motion for isotropic Newtonian fluids is investigated. The fluids are compressible, viscous and heat-conductive. It is proved that there exists a unique global solution in time, for the small initial data, and the solution has the decay rate of (1 + t) to 3/4 power as t approaches positive infinity. The motions of compressible, viscous and heat-conductive fluids are described by a system of partial differential equations which is of hyperbolic-parabolic type and highly nonlinear. One of the first mathematical problems associated with this system is the initial value problem. We obtain the existence of a a unique smooth global solution in time for the initial value problem and also the decay rate of the solution as time tends to infinity.

Stochastic behavior of a quantum pendulum under a periodic perturbation

Abstract: This paper discusses a numerical technique for computing the quantum solutions of a driver pendulum governed by the Hamiltonian $$H = (p_\theta ^2 /2m\ell ^2 ) - [m\ell ^2 \omega _o ^2 \cos \theta ] \delta _p (t/T) ,$$ where pe is angular momentum, ϑ is angular displacement, m is pendulum mass, l is pendulum length, ωO2 = g/l is the small displacement natural frequency, and where δp (t/T) is a periodic delta function of period T. The virtue of this rather singular Hamiltonian system is that both its classical and quantum equations of motion can be reduced to mappings which can be iterated numerically and that, under suitable circumstances, the motion for this system can be wildly chaotic. Indeed, the classical version of this model is known to exhibit certain types of stochastic behavior, and we here seek to verify that similar behavior occurs in the quantum description. In particular, we present evidence that the quantum motion can yield a linear (diffusive-like) growth of average pendulum energy with time and an angular momentum probability distribution which is a time-dependent Gaussian just as does the classical motion. However, there are several surprising distinctions between the classical and quantum motions which are discussed herein.

The evolution of the lunar orbit revisited. I

Abstract: After recalling the contribution of Halley, J. Kepler, and G. Darwin to our understanding of the secular acceleration of the Moon, we establish a set of differential equations for the variation of the semi-major axis, and the inclination of the Moon on the maximum area plane. These equations are obtained without expanding the disturbing function, due to the tidal bulge, in term of the elliptic elements. The equations thus obtained are simple enough to allow us a qualitative discussion of the solution, followed by a numerical integration. The results obtained show the Moon was in the distant past in a retrograde orbit, approaching the Earth, its inclination increasing towards 90°; once after a closer approach to the Earth, the Moon receeded and it will finally reach an equilibrium point, the orbital and the equatorial planes being blended. The solution of the equations appears as a fascicle of curves, becoming extremely dense as we come nearer to the present. Owing to the high sensitivity of the solution to the initial conditions, a weak disturbance added to our modeled forces may lead to a past situation very different from the conclusion drawn by Goldreich (1966) and MacDonald (1964); the minimal approach distance could be greater than 10 Earth's radii.

Motion of three vortices

Abstract: A qualitative analysis of the motion of three point vortices with arbitrary strengths is given. This simplifies and extends recent work by Novikov on the motion of three identical vortices. Using a phase diagram technique, the possible regimes of motion are classified according to the signs of the arithmetic, geometric, and harmonic means of the three vortex strengths. For the special case where the vortex strengths (κ1,κ2,κ3) take the values (+κ,+κ,−κ), the diagram has an interpretation in terms of the scattering of a neutral pair by a single vortex. Quantitative details are presented for this case. If the harmonic mean of the three vortex strengths is zero, the triangle of vortices can collapse to a point in a finite time for certain initial conditions.

The Theory of Ship Motions

Abstract: Publisher Summary This chapter highlights that the oceangoing ships are designed to operate in a wave environment that is frequently uncomfortable and sometimes hostile Unsteady motions and structural loading of the ship hull are two of the principal engineering problems that result Ships generally move with a mean forward velocity and their oscillatory motions in waves are superposed upon a steady flow field The solution of the steady-state problem is itself of interest, particularly with regard to the calculation of wave resistance in calm water The problem of ship motions in waves can be regarded as a superposition of these two special cases, but interactions between the steady and oscillatory flow fields complicate the more general problem The chapter also discusses the dynamics of ship motions that are governed by the equations of motion that balance the external forces and moments acting upon the ship, with the internal force and moment because of gravity and inertia Assuming the ship to be in stable equilibrium in calm water, its weight is balanced by the force of hydrostatic pressure Similarly, the steady drag and propulsive force are balanced These steady forces may be neglected and attention is focused on the unsteady perturbations

A guiding center Hamiltonian: A new approach

Abstract: A Hamiltonian treatment of the guiding center problem is given which employs noncanonical coordinates in phase space. Separation of the unperturbed system from the perturbation is achieved by using a coordinate transformation suggested by a theorem of Darboux. As a model to illustrate the method, motion in the magnetic field B=B (x,y) ? is studied. Lie transforms are used to carry out the perturbation expansion.

Kink-Antikink Collisions in the Two-Dimensional φ4 Model

Abstract: Inelastic kink-antikink collisions are investigated in the two-dimensional ¢' model. It is shown that a bound state of the kink and antikink is formed when the colliding velocity V is less than a critical V" and that inelastic scattering of the kinks is caused when V> V,. These phenomena are due to the excitation of internal modes of the kinks. An excited kink is not stable but emits bosuns to decay into the ground state kink. Numerical calculations were also performed for classical kink-antikink collision processes. A large class of non-linear field equations has localized stable soh;tions which are called solitons. The quantum theory of soliton has been studied by many authors aiming at constructing a model of extended particles. Semi-classical meth­ ods have been applied to the quantization of a soliton and the systematic pertur­ bation theories have been developed at least in one-soliton sector.n' 2 ' However, we have no satisfactory treatment of the quantum scattering of solitons. 3l In the strict meaning used in mathematical literature, solitons are stable even against collisions. Since this stability is guaranteed by an infinite number of con­ served quantities,·') these solitons are scattered elastically also in quantum theory. The sine-Gordon model provides a good example of such solitons. In particle physics, however, it may be useful to understand solitons in a broad sense. That is, if a field equation has a classical solution vvhose energy is localized in a finite volume and which is stable against small fiuctional variations, we call it a soliton solution. In this sense, there are solitons which are unstable against a collision, and it is important to study inelastic scattering of such solitons. In this paper, we investigate soliton-antisoliton collision processes m the two­ dimensional ¢4 model as an example of the inelastic scattering of solitons. The two-dimensional ¢4 model has the so-called kink solutions and a fluctuation mode trapped in the kink. The kinks are scattered inelastically because of the excitation of the internal modes. In §§ 2 and 3, we introduce a collective coordinate into a kink-antikink system and derive the equations of motion for the system. Solving the equations approximately, we estimate the excitation probability of the internal

The motion of small particles in non-Newtonian fluids

Abstract: There are many problems in which the motion of a small particle, bubble or drop in a non-Newtonian fluid differs in an important qualitative way from its corresponding motion in a Newtonian fluid. From a theoretical point of view such problems are conveniently separated into two groups. In the first, some aspect of the particle's motion only exists, for small Reynolds number, because the suspending fluid is non-Newtonian. Examples of this class include the cross-stream (or lateral) motion of spherical particles in a unidirectional shear flow, rotational motion of an orthotropic particle in sedimentation (leading to a deterministic equilibrium orientation), and cross-orbital drift in the rotation of an axisymmetric particle in shear flow. In these cases, a major change in the orientation or position of the particle can result from small instantaneous contributions of non-Newtonian rheology to the particle's motion, provided that these act “cumulatively” over a sufficiently long period of time. An analytical description of the fluid mechanics relevant to this process may thus be based on the asymptotic limit of a nearly-Newtonian fluid using the so-called “retarded-motion” expansion, and a relevant constitutive model for viscoelastic materials is the Rivlin—Ericksen nth-order fluid. Comparison between theory and experiment shows excellent qualitative (and frequently quantitative) agreement for such problems even when the flow is too rapid, in a rheological sense, for strict adherence to the requirements of a retarded-motion expansion. The second major class of problems is that in which the observed difference between Newtonian and non-Newtonian behavior is due to an important, O(1) change in the fluid motion at all times. In this case, the only possible theoretical description which is valid in more than an asymptotic sense is one based on a full non-linear constitutive model, including “memory”, and thus a solution of the equations of motion is generally possible only via numerical methods. Unlike the first class of problems, an important determining factor in successful match between experiment and theory is therefore a judicious (fortunate?) choice of the constitutive model. In the second part of this paper, I shall discuss some examples of numerical and experimental studies which pertain to particle motions in the regime of strongly viscoelastic flows.

DTNSRDC Revised Standarrd Submarine Equations of Motion

Abstract: : Revised standard equations of motion are presented for use in computer simulations of submarine motions in six degrees of freedom conducted for the US Navy. (Author)

Adiabatic contraction of anisotropic spheres in general relativity

Abstract: For specific equations of state we study the adiabatic contraction of a sphere with anisotropic stresses for different degrees of anisotropy. The equation of motion of particles in the matter is integrated numerically for different initial conditions. It is found that the local pressure-density law for equilibrium is different for different regions in the material. In some examples the models are more stable than the isotropic model; in others they are less stable.

Particle Dynamics in the Plasma Sheet

Abstract: Trajectories of charged particles in the trail region of the earth's magnetosphere are studied numerically using a model magnetic field. It is shown that both trapped and untrapped trajectories can be categorized into three groups using two dimensionless parameters. One of the parameters is the ratio of the x and z components of the model magnetic field. The other is the ratio of the plasma sheet thickness to the particle gyroradius in the midplane. Previous trajectory studies are incorporated into our classification scheme which resolves a number of apparent contradictory conclusions among them. The untrapped particles are analyzed in terms of the cross-tail displacement, the displacement time, the scattering of the final pitch angle, and the coefficient of reflection off the plasma sheet. The results are used to predict the upper limit on particle energization during one neutral sheet interaction. An understanding of the motions of individual charged particles in the magnetotail plasma sheet is an important first step in understanding the collective dynamics of the magnetotail plasma as a whole. A number of authors have attempted analytical solutions of the equations of motion by various approximations. Among them Speiser [1965], Alexeev and Kropotkin [ 1970], and Sonnerup [ 1971 ] have obtained approximate solutions in the highly nonadiabatic limit. On the other hand, Stern and Palmadesso [1975] and Stern [1977] have obtained approximate solutions in the adiabatic limit. Other authors have attempted to numerically integrate the equations of motion, for example, Speiser [1967], Cowley [1971], Eastwood [1972], Pudovkin and Tsyganenko [1973], and Swift [1977]. However, due to the assumptions made in their studies, each author dealt only with a limited variety of trajectories, and therefore their conclusions cannot be generalized to all conditions that occur in the magentotail. In this paper, a numerical analysis of all possible magnetotail particle orbits is presented. The previous works are incorporated into a new classification scheme, which resolves the apparent disagreements among a number of previous studies. The new classification makes use of the dimensionless equations of motion suggested by Swift [1977]. The overall morphology of the orbits is shown to be determined by two dimensionless parameters in the equations of motion. 2. THE MODEL AND THE EQUATIONS OF MOTION To analyze the motion of a charged particle in the plasma sheet, we chose a magnetic field which is given by

Flow through tubes with sinusoidal axial variations in diameter

Abstract: An iteration technique has been developed to solve the equations of motion for flow of an incompressible Newtonian fluid through circular tube with a radius which varies sinusoidally in the axial direction. The iteration is essentially geometric; one proceeds from a solution for flow through a tube in which the waveleneth of diameter chance in the axial direction is

Discrete Symmetric Dynamical Systems at the Main Resonances with Applications to Axi-Symmetric Galaxies

Abstract: A study of two-degrees-of-freedom systems with a potential which is discrete-symmetric (even in one of the position variables) is carried out for the resonance cases 1:2, 1:1, 2:1 and 1:3. To produce both qualitative and quantitative results, we obtain in each resonance case normal forms by higher order averaging procedures. This method is related to Birkhoff normalization and provides us with rigorous asymptotic estimates for the approximate solutions. The normal forms have been used to obtain a classification of possible local and global bifurcations for these dynamical systems. One of the applications here is to describe the two-parameter family of bifurcations obtained by detuning a one-parameter family studied by Braun. In all the resonances discussed an approximate integral of the motion other than the total energy exists, but in the 2:1 and 1:3 resonance cases this degenerates into the partial energy of the z motion. In conclusion some remarks are made on the relation between two-degrees-of-freedom systems and solutions of the collisionless Boltzmann equation. Moreover we are able to make some observations on the Henon-Heiles problem and certain classical examples of potentials.

Normal form for mirror machine Hamiltonians

Abstract: A systematic algorithm is developed for performing canonical transformations on Hamiltonians which govern particle motion in magnetic mirror machines. These transformations are performed in such a way that the new Hamiltonian has a particularly simple normal form. From this form it is possible to compute analytic expressions for gyro and bounce frequencies. In addition, it is possible to obtain arbitrarily high order terms in the adiabatic magnetic moment expansion. The algorithm makes use of Lie series, is an extension of Birkhoff’s normal form method, and has been explicitly implemented by a digital computer programmed to perform the required algebraic manipulations. Application is made to particle motion in a magnetic dipole field and to a simple mirror system. Bounce frequencies and locations of periodic orbits are obtained and compared with numerical computations. Both mirror systems are shown to be insoluble, i.e., trajectories are not confined to analytic hypersurfaces, there is no analytic third i...

Noether’s theorem, time‐dependent invariants and nonlinear equations of motion

Abstract: Noether’s theorem is applied to a Lagrangian for a system with nonlinear equations of motion. Noether’s theorem leads to a time‐dependent constant of the motion along with an auxiliary equation of motion. Special cases of this invariant have been used to quantize the time‐dependent harmonic oscillator. We also discuss the solution of the original equations of motion in terms of the solutions to the auxiliary equation.

The principle of virtual work and integral laws of motion

Abstract: This paper furnishes a simple constructive proof of the equivalence of integral laws of motion for continua to the Principle of Virtual Work. The approach used is designed to avoid the artificiality of introducing the classical equations of motion in an intermediate step. The hypotheses employed are virtually the weakest possible that are consistent with the requirement that the integrals appearing in these formulations make sense as Lebesgue integrals. Particular attention is devoted to the treatment of boundary conditions, which may assume a very general form.

Reduction of the two‐dimensional O(n) nonlinear σ‐model

Abstract: We reduce the field equations of the two‐dimensional O(n) nonlinear σ‐model to relativistic O(n−2) covariant differential equations involving n−2 scalar fields.

Lagrangians and Systems They Describe-How Not to Treat Dissipation in Quantum Mechanics.

Abstract: A Lagrangian that yields equations of motion for a damped simple harmonic oscillator is shown not to describe this system, but a completely different physical system. Thus, we have a simple counterexample to the statement: ’’If a Lagrangian gives the correct equations of motion for a given system then the Lagrangian describes the system.’’ We also construct a physical system that the Lagrangian describes and derive some of its properties. The quantum theory based on the above Lagrangian has been used to attempt to discuss the quantum theory of dissipation. This is, however, incorrect since the system the Lagrangian describes has no dissipation.

Applications of Generalized Derivatives to Viscoelasticity.

Abstract: : Generalized derivatives of fractional order are used to construct stress-strain constitutive relations for viscoelastic materials, based on the observed sinusoidal behavior of the materials The non-periodic behavior of one material is observed in the laboratory and compares favorably with the non-periodic behavior of the material predicted by its generalized derivative constitutive relation Having established that the generalized derivative constitutive relation is an appropriate mathematical model for the general motion of at least one viscoelastic material, the tools for the analysis of structures of engineering interest are put forward In particular, attention is focused on a finite element formulation of and solutions to the equations of motion for structures containing elastic and viscoelastic components (Author)

Asymptotic Behaviors in the Dynamics of Competing Species

Abstract: We consider systems of interacting species obeying Lotka–Volterra equations. We show that periodic attractors may be generated from the equilibrium point in phase space by Hopf bifurcation. An exception is the case of three species systems with equal individual growth rates, where Hopf bifurcation is abnormal. We show that the dynamics of such systems is actually bidimensional, a fact which permits us to give an analytic description of the closed orbits observed by May and Leonard. We also consider the “explosive” dynamics of a system without self interactions and show that adding a constant source term in the equations of motion has a stabilizing effect, producing periodic solutions. In the limit of small external source terms we are able to describe analytically the associated cyclic evolutions. This effect of source terms on the solutions of Lotka–Volterra equations seems to be quite general, and is explained by the fact that they remove the attraction of the orbits by the coordinate planes.

Coupled inertial and gravitational effects in the proper reference frame of an accelerated, rotating observer

Abstract: To the second order in metric and the first order in equations of motion in the local coordinates of an accelerated rotating observer, the inertial effects and gravitational effects are simply additive. To look into the coupled inertial and gravitational effects, we derive the third‐order expansion of the metric and the second‐order expansion of the equations of motion in local coordinates. Besides purely gravitational (purely curvature) effects, the equations of motion contain, in this order, the following coupled inertial and gravitational effects: redshift corrections to electric, magnetic, and double‐magnetic type curvature forces; velocity‐induced special relativistic corrections; and electric, magnetic, and double‐magnetic type coupled inertial and gravitational forces. An example is provided with a static observer in the Schwarzchild spacetime.