Classical Dynamical Equations Research Articles

Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrodinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrodinger-Boussinesq system and the Schrodinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Henon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Henon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Henon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.

The general equations of analytical dynamics

It is shown that the generalized Poincaré and Chetayev equations, which represent the equations of motion of mechanical systems using a certain closed system of infinitesimal linear operators, are related to the fundamental equations of analytical dynamics. Equations are derived in quasi-coordinates for the case of redundant variables; it is shown that when an energy integral exists the operator X 0 = ∂/ ∂t satisfies the Chetayev cyclic-displacement conditions. Using the energy integral the order of the system of equations of motion is reduced, and generalized Jacobi-Whittaker equations are derived from the Chetayev equations. It is shown that the Poincaré-Chetayev equations are equivalent to a number of equations of motion of non-holonomic systems, in particular, the Maggi, Volterra, Kane, and so on, equations. On the basis of these, and also of other previously obtained results, the Poincaré and Chetayev equations in redundant variables, applicable both to holonomic and non-holonomic systems, can be regarded as general equations of classical dynamics, equivalent to the well-known fundamental forms of the equations of motion, a number of which follow as special cases from the Poincaré and Chetayev equations.

The Heisenberg equation for phonon operators and soliton solutions in nonlinear lattices

The Heisenberg equation for phonon operators in nonlinear lattices is derived establishing the interaction Hamiltonian included higher powers of particle-hole pairs in nonlinear lattices. A phonon operator consists of a particle-hole pair in the harmonic potential approximation in the two band model; it represents an up or down transition of atoms between two levels. Applying the boson transformation method to the Heisenberg equation for phonon operators, we obtain the classical dynamical equation and a linear equation with the self-consistent potential created by the extended objects in nonlinear lattices. The boson transformation leads to soliton solutions in the long wavelength limit. The linear equation can be used to obtain scattering states, bound states and translational modes for phonons.

Mathematical theory of the origin of matter

It is shown that from a simple and elegant action it is possible to obtain: (i) the equations of classical dynamics, (ii) Schrodinger's equation, (iii) the dynamical equation of special relativity, (iv) scale-invariant gravitation including general relativity, (v) the mathematical theory of the origin of matter, and (vi) the potential function of inflationary theory. When the action term in question is related to the electromagnetic theory an ugly feature arises, however. There must be a multiplication by a small dimensionless number of order 10−38. If this ugly feature is to be avoided, matter must be taken to originate, not as particles observed in the laboratory but as Planck particles. The decay of each such particle into ≈ 1019 hadrons then explains the genesis of numbers of order 1038 that appear in physics and cosmology. It also raises questions concerning primordial nucleosynthesis which were discussed in the preceding paper.

Measurement of total urethral compliance in females with stress incontinence.

Urodynamic evaluation of stress incontinence has failed to result in consistent diagnostic parameters. Using retrospective female bladder pressure and urine flow data, we evaluated two parameters based on a conductance formula. This formula was derived from fitting a classical fluid dynamic equation to published data of voiding cystourethography. The Area Equivalent Factor-female at maximal flow represents maximal urethral opening and this value further divided by the pressure is the total urethral compliance. A stress incontinent group had a strong trend toward greater maximal urethral opening and total urethral compliance than a continent group. These parameters may, therefore, have potential in the evaluation and understanding of stress urinary incontinence.

Characterization of the thermal behaviour in ICs

A complete four-dimensional solution T( x, y, z, t) to the classical dynamic heat-flow equation, for the case where the power sources lie on the silicon surface, is presented. The solution is found by: (i) applying multiple finite Fourier transforms with respect to x and y; (ii) Laplace-transforming with respect to time; (iii) solving the resulting differential equations for the z variable and determining the Fourier components; (iv) performing the inverse Laplace transforms for all the Fourier terms and summing them. The solution has been used to simulate the thermal behaviour of several different IC power configurations.

An equation for acoustic propagation in inhomogeneous media with relaxation losses

An equation for acoustic propagation in an inhomogeneous liquid medium with relaxation loss is systematically derived from the classical dynamic equations together with the appropriate thermodynamic and constitutive relations. The derivation assumes small acoustic perturbations, but accommodates arbitrary spatial inhomogeneities in material compressibility, density, and parameters of relaxation. The linearized wave equation obtained for N relaxation mechanisms has order N+2, is causal, and yields the expected dependency of attenuation and phase velocity on frequency. A similar equation is also obtained for gases. The corresponding reduced wave equation simplifies to the classical form with a complex frequency-dependent compressibility for which the Kramer–Kronig conditions are verified and for which an explicit expression is obtained in terms of medium properties. Exact analytic expressions valid at all frequencies are given for the spatially varying attenuation coefficient as well as phase velocity. A Green’s function is found for the constant coefficient part of the equation. The results are used to illustrate the effect of relaxation on scattering and also may be employed to model scattering in applications such as image reconstruction.

Topology of quadratic super-Hamiltonians

Dynamical systems with finite number of degrees of freedom and one constraint are considered; this constraint, called a super-Hamiltonian, is assumed to be a quadratic function of all canonical coordinates and momenta. The authors examine those topological properties of the orbits generated by the super-Hamiltonians that are relevant to the existence of cross-sections (Gribov problem). The well known results on the so-called real normal form of quadratic Hamiltonian functions are extended to super-Hamiltonians and briefly summarised. They are used to separate the classical dynamical equation. The dynamics in the separation subspaces helps to find the topology of the orbits and to give its complete description. One of the results is that the number of generic super-Hamiltonian families that do not admit cross-sections grows with the number of degrees of freedom much more rapidly than the number of families that do. Thus, orbit topology that does not admit cross sections is not just a marginal, pathological effect.

Can classical equations simulate quantum‐mechanical behavior? a molecular dynamics investigation of a diatomic molecule with a morse potential

AbstractIn a recent paper, we presented a new computational method for molecular dynamics which uses the Backward‐Euler scheme to solve the classical Langevin dynamics equations. Parameters for the simulation include a target temperature T, a time step Δt, and a cutoff frequency ωc. We showed for a harmonic oscillator system that the cutoff frequency can be set as ωc = kT/h in order to mimic quantum‐mechanical behavior. We now continue this investigation for a nonlinear case: a diatomic molecule governed by a Morse bond potential. Since approximate quantum‐mechanical energy levels are explicitly known for this model, a comparison of energies can be made with molecular dynamics results. By performing dynamics runs for a wide range of temperatures and calculating mean energies, we find a very good agreement between these energies and quantum mechanical predictions. Vibrational excitation begins at temperatures around 800 K, and for higher temperatures both energy curves (molecular dynamics and quantum mechanics) approach the classical prediction of 7/2kT energy per molecule. Future investigations will focus on more general nonlinear potential functions employed in force fields of nucleic acids and proteins.

An equation for acoustic propagation in an inhomogeneous medium with relaxation loss

An equation for acoustic propagation in an inhomogeneous medium with relaxation loss is systematically derived from the classical dynamic equations together with an equation of state for relaxation. The derivation assumes small acoustic perturbations but accommodates arbitrary spatial inhomogeneities in material compressibility, density, and parameters of relaxation. The linearized wave equation obtained for n relaxation mechanisms has order n + 2, is causal, and yields the expected dependence of attenuation on frequency. Exact analytic expressions valid at all frequencies are given for the spatially varying attenuation coefficient, as well as phase velocity. A Green's function is calculated for the equation. The results may be used to model scattering for image reconstruction and the determination of statistical properties, such as average differential scattering cross section.

An alternative classical cosmological model with inflationary expansion

We propose a cosmological model of the Universe based on the Newtonian mechanics and classical field theory. The essential ingredient of this model is the existence of a special kind of physical field in the Universe whose source is the mass current. In the early Universe this field reached such large values that it produced matter from the vacuum fluctuation. The classical dynamical equations for the co-moving sphere in the presence of this field are enlarged by a new term which causes an inflation-like expansion. It accounts also for the hot initial stage of the early Universe and has several important cosmological consequences.

Quantization of dynamical systems subject to reducible second-class constraints

A universal definition is given for irreducible and any-stage-reducible second-class constraints and corresponding modified Dirac brackets. The classical dynamics equations are represented in terms of the modified Dirac brackets and the classical Cauchy problem is shown to be locally nondegenerate. The general and the special solution for theS-matrix is constructed for dynamical systems subject to second-class constraints of any stage of reducbility.

A molecular theory of the solid–liquid interface. II. Study of bcc crystal–melt interfaces

We have extended and applied our previously developed order parameter theory for the solid–liquid interface [J. Chem. Phys. 74, 2559 (1981)]. We show how the differential equations for interfacial structure from our earlier work may be reformulated in terms of a variational principle and demonstrate that the quantity minimized gives the surface free energy. We point out a useful analogy between the order parameter equations and the equations of classical dynamics. We carry out explicit calculations of the interfacial structure of the bcc 100 and 111 crystal–melt interfaces and obtain estimates of the solid–liquid interfacial free energy for sodium and potassium; the free energies differ by less than 3% for the two different faces of each crystal. Our calculations suggest the existence of a ’’structured liquid’’ region in the interface, characterized by a liquidlike average density, but solidlike lattice ordering. The interfacial transition zone is quite broad, 16–20 Å in width and comprising 10–15 layers. We compare our results to computer simulation data and discuss their implications for microscopic mechanisms of crystal nucleation and growth from the melt.

Time-dependent quadratic constants of motion, symmetries, and orbit equations for classical particle dynamical systems with time-dependent Kepler potentials

It is shown there are only two classes of time-dependent Kepler potentials [V2≡λ0(at2+bt+c)−1/2/r, (b2−4ac≠0), and V3≡λ0(αt+β)−1/r] for which the associated classical dynamical equations will admit quadratic first integrals more general than quadratic functions of the angular momentum. In addition to the angular momentum the system defined by V2 admits only a ’’generalized time-dependent energy integral,’’ while the system defined by V3 admits in addition to these a time-dependent vector first integral that is a generalization of the Laplace–Runge–Lenz vector constant of motion (associated with the time-independent Kepler system). For the V3 system the time-dependent vector first integral is employed to obtain in a simple manner the orbit equations in completely integrated form. The complete group of (velocity-independent) symmetry mappings is obtained for each of these two classes of dynamical systems and used to show that the generalized energy integral is expressible as a Noether constant of motion.

A model of sulfate aerosol dynamics in atmospheric plumes

The evolution of atmospheric aerosols is currently a subject of concern because of its relationship to environmental issues. Recently, a rigorous approach for modeling the dynamics of a sectional aerosol distribution has been developed by Gelbard, Tambour and Seinfeld (1980) J. Colloid Interface Sci., 76, 541–556. This paper makes use of this approach and extends it through the presentation of a mathematical model that describes advective transport, turbulent diffusion, gas-phase chemistry and aerosol dynamics in atmospheric plumes. This mathematical model incorporates the Reactive Plume Model, which is a Lagrangian model used to describe plume dynamics consisting of six contiguous cells that expand as the plume is dispersed by atmospheric turbulence. It assumes a Gaussian distribution of the plume arid takes into account interactions with the ambient air. The mathematical model also incorporates the Carbon-Bond Mechanism to model the gas-phase chemistry, which involves 73 reactions among 36 chemical specks. The aerosol population distribution for this model is represented by means of seven sections that have a particle diameter range of 0.01–2.15 μm. Equations describing aerosol dynamics are derived from the classical General Dynamic Equation (CDE) for a sectional aerosol distribution. The dynamic processes described in the model include inter- and intra-sectional coagulation, condensation of acid sulfate or ammonium sulfate and heterogeneous oxidation of sulfur dioxide (SO 2) in each section, as well as new-particle formation in the lowest section. Condensation is diffusion-limited and new-particle formation is estimated from monomer balance theory. Heterogeneous oxidation of SO 2 in the aerosol phase is parameterized by means of a pseudo-first-order chemical reaction. The aerosol dynamics module is evaluated with smog chamber data obtained for an aerosol-H 2SO 4-H 2O system. Aerosol plume model predictions are compared with airborne measurements obtained from the plume of the Navajo power plant on four different days during the VISTTA program, at downwind distances ranging from 25 to 88km. Important characteristics of aerosol dynamics are well described by the model. Secondary aerosol formation is most apparent in the 0.01–0.1 μm size range. Maximum predicted SO 2 homogeneous oxidation rates are 0.5 and 0.6 % h −1 for the summer and winter cases, respectively. Sensitivity studies performed with the aerosol plume model showed that background hydrocarbon concentrations, relative humidity and photolysis rates affect secondary aerosol formation most, whereas atmospheric stability has little effect.

Stellar wind theory

The theory of stellar winds as given by the equations of classical fluid dynamics is considered. The equations of momentum and energy describing a steady, spherically symmetric, heat‐conducting, viscous stellar wind are cast in a dimensionless form which involves a thermal conduction parameter E and a viscosity parameter γ. An asymptotic analysis is carried out, for fixed γ, in the cases E → 0 and E → ∞ (corresponding to small and large thermal conductivity, respectively), and it is found that it is possible to construct critical solutions for the wind velocity and temperature over the entire flow. The E → 0 solution represents a wind which emanates from the star at low, subsonic speeds, accelerates through a sonic point, and then approaches a constant asymptotic speed, with its temperature varying as r−4/3 at large distances r from the star; the E → ∞ solution represents a wind which, after reaching an approximately constant speed, with temperature varying as r−2/7, decelerates through a diffuse shock and approaches a finite pressure at infinity. A categorization is made of all critical stellar wind solutions for given values of γ and E, and actual numerical examples are given. Numerical solutions are obtained by integrating upstream ‘from infinity’ from initial values of the flow parameters given by appropriate asymptotic expansions. The role of viscosity in stellar wind theory is discussed, viscous and inviscid stellar wind solutions are compared, and it is suggested that with certain limitations, the theory presented may be useful in analyzing winds from solar‐type stars.

Dynamics and ensemble averages for the polarization models of molecular interactions

A family of ’’polarization models’’ has been generated for description of polarizable and deformable molecules in condensed phases. Within these models the intermolecular forces are not pairwise additive, but have a many-body component that generalizes the polarization interactions present in conventional electrostatics. It is shown that the classical dynamical equations have a rather compact form, in spite of the many-body interactions. Novel molecular distribution functions are introduced in terms of which the usual formulas of statistical thermodynamics can be expressed. Finally, the static dielectric constant is discussed for the polarization models, and it is related to fluctuating local moments in the system.

Molecular dynamics and chemical reactivity

A computer simulation has been obtained of the atom recombination reaction in which the recombination energy is removed by a third-body. The equations of classical dynamics have been solved for two iodine atoms and six inert gas atoms (He, Ar or Xe) confined to a spherical vessel by a wall potential which is a spatial average of a spherical shell of inert gas atoms. Reactions at a given temperature and concentration are simulated by varying the initial momenta of the atoms and the volume of the sphere. A computer run of 100 trajectories for each physical situation gives the average time for recombination, and from this a macroscopic rate constant has been calculated that agrees well with the experimental result. The model reproduces all the characteristic kinetic mechanisms that have traditionally been used to interpret atom recombination. However, at high inert gas concentrations the steady-state approximation is shown to fail as many of the important intermediate reactions do not reach equilibrium. In t...

Topics in computational fluid mechanics

The partial differential equations of classical fluid dynamics describe a non-linear continuum with an infinite number of degrees of freedom, which must be simulated on the computer by some form of discrete representation. A brief survey is given of some of the many different types of flow that can occur, indicating how the analytic and numerical techniques that are found to be most appropriate in practice can be related to particular physical characteristics of the problem in hand. Computer simulations of incompressible flows are of especial interest since stable non-linear structures such as vortices are often observed to arise from initially unstable situations. Computer movies of 2-dimensional incompressible flow were shown during the talk to demonstrate this process of vortex condensation, which can be studied in detail by means of controlled numerical experiments. A qualitative explanation is available in terms of hamiltonian statistical mechanics but much further work remains to be done.

Quantum Theory of Superfluid Vortices. I. Liquid Helium II

A Hamiltonian formulation is constructed for the classical dynamical equations of slightly deformed rectilinear vortices. The system is then quantized by interpreting the conjugate variables as quantum-mechanical operators that obey canonical commutation relations. A linear canonical transformation diagonalizes the Hamiltonian in terms of operators that create and destroy single quanta of vortex vibrations. The theory is applied to two distinct configurations in He II: a single vortex and a rotating vortex lattice. The specific heat associated with the vortex waves varies approximately as ${T}^{\frac{1}{2}}$ at low temperatures. Quantum-mechanical and thermal fluctuations produce a finite mean-square displacement of the vortex core, which is studied both at $T=0$ and at $Tg0$.