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DOE OSTI2026

A Structure-Preserving Decorated Particle Method for the Vlasov-Poisson System

We revisit the Scovel-Weinstein framework (Scovel & Weinstein, CPAM 1994) for reducing the Vlasov-Poisson system while preserving its Hamiltonian structure. Standard particle-in-cell (PIC) algorithms approximate the distribution function by macro-particles with position and velocity. In contrast, Scovel-Weinstein decorated particles involve additional shape degrees of freedom, while maintaining a finite-dimensional reduction with Hamiltonian structure inherited from the continuum model. Although the original work established this structure three decades ago, its computational potential has remained largely unexplored. We present a practical implementation of the Scovel-Weinstein model and compare it with a standard PIC algorithm. Numerical experiments demonstrate that macro-particles in standard PIC can be replaced by far fewer decorated particles while retaining comparable accuracy. This decorated particle approach offers a new structure-preserving paradigm for kinetic plasma simulation.

65M75, 70H05, 70G65↗

DOE OSTI2024

Microscopic Derivation of Transition-state Theory for Complex Quantum Systems

The decay of quantum complex systems through a potential barrier is often described with transition-state theory, also known as RRKM theory in chemistry. Here we derive the basic formula for transition-state theory based on a generic Hamiltonian as might be constructed in a configuration-interaction basis. Two reservoirs of random Hamiltonians from Gaussian orthogonal ensembles are coupled to intermediate states representing the transition states at a barrier. Under the condition that the decay of the reservoirs to open channels is large, an analytic formula for reaction rates is derived. Here, the transition states act as independent Breit–Wigner resonances which contribute additively to the total transition probability, as is well known for electronic conductance through resonant tunneling states. It is also found that the transition probability is independent of the decay properties of the states in the second reservoir over a wide range of decay widths.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

NASA NTRS1965

Studies in Perturbation Theory. XI. Lower Bounds to Energy Eigenvalues, Ground State, and Excited States

The bracketing theorem in the partitioning technique for solving the Schrödinger equation may be used in principle to determine upper and lower bounds to energy eigenvalues. Practical lower bounds of any accuracy desired may be evaluated by utilizing the properties of ``inner projections'' on finite manifolds in the Hilbert space. The method is here applied to the ground state and excited states of a Hamiltonian H=H(sub 0)+V having a positive definite perturbation V. Even if inspiration is derived from the method of intermediate Hamiltonians, the final results are of bracketing type and independent of this approach. The method is numerically illustrated in some accompanying papers.

Loewdin, Per-Olov↗

NASA NTRS1972

The stability of motion of satellites with flexible appendages

The mathematical formulation associated with the problem of stability of motion of a satellite consisting of a main rigid body and three (or less) pairs of flexible rods is presented. The rods are capable of flexure in two orthogonal directions. Whereas the rotational motion of the body is described by generalized coordinates depending on time alone, the elastic displacements of the rods depend both on spatial position and time. Assuming no external torques, there exist motion integrals in the form of momentum integrals. These integrals can be regarded as constraint equations relating the system velocities, and used to reduce the number of variables describing the motion. The stability analysis has been carried out by means of an extension of the Liapunov direct method. Since the elastic vibrations result in energy dissipation, it is shown that the equilibrium position is asymptotically stable if the Hamiltonian is positive definite and unstable if it can take negative values in the neighborhood of the equilibrium. Determining the sign definiteness of the Hamiltonian is complicated by the fact that it contains spatial derivatives of the elastic displacements. Two methods are presented to cope with this problem. The first, the standard modal analysis in conjunction with series truncation, develops criteria in terms of infinite series associated with the natural modes and frequencies of the elastic rods. The second, the method of integral coordinates, yields closed-form stability criteria involving the system parameters, such as the body moments of inertia, the length and mass distribution of the rods, the lowest natural frequencies of the rods, and the satellite spin velocity.

Meirovitch, L.↗

NASA NTRS1972

The method of models in variational scattering calculations.

It is shown that difficulties in atomic scattering calculations that stem from the use of inexact target wave functions can be overcome by employing the so-called method of models, provided that the projectile is distinguishable from the atomic electrons. The proposed method of models consists in replacing the target hamiltonian by a model hamiltonian, of which the appropriate target state is an eigenfunction. A connection with the positron scattering work of Peterkop and Rabik (1971) has been established.

Drachman, R. J.↗

NASA NTRS1972

A simple, radially correlated ground state wavefunction for two electron atoms.

A one parameter function is presented as an approximation to the ground state wavefunction of the two electron radial hamiltonian. The parameter may be fixed by a nonvariational criterion. The resulting expectation value of the radial hamiltonian differs from its exact eigenvalue by about 2 parts in 3000 for helium while the 'local energy' never differs by more than 10% from the exact value over the entire r1-r2 plane. The cases Z = 1 and Z = 3 are also investigated.

Altick, P. L.↗

NASA NTRS1972

Effects of gravity-gradient torque on the rotational motion of a triaxial satellite in a precessing elliptic orbit.

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the effects of gravity-gradient torque on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is free to precess and spin. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite. The differential equations derivable from the transformed Hamiltonian are integrable, and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed, and a comparison of theoretical results with observations is made.

Cochran, J. E.↗

NASA NTRS1973

Liapunov stability analysis of spinning flexible spacecraft.

The attitude stability of a class of spinning flexible spacecraft in a force-free environment is analyzed. The spacecraft is modeled as a rigid core having attached to it a flexible appendage idealized as a collection of elastically interconnected particles. Liapunov stability theorems are employed with the Hamiltonian of the system, constrained through the angular momentum integral so as to admit complete damping, used as a testing function. The Hamiltonian is written in terms of modal coordinates as interpreted by the hybrid coordinate formulation, thus allowing truncation to a level amenable to literal stability analysis. Testing functions are constructed for a spacecraft with an arbitrary (discretized) appendage, and closed form stability criteria are generated for the first mode of a restricted appendage model lying in a plane which contains the center of mass and is orthogonal to the spin axis. The criteria are (except for idealized cases on the stability boundary line in the parameter space) both necessary and sufficient for stability for any spacecraft characterized by the planar appendage model, such as a spacecraft containing solar panels and/or radial booms.

Barbera, F. J.↗

NASA NTRS1973

Long-term motion in a restricted problem of rotational motion

A method of general perturbations, based on the use of Lie series to generate approximate canonical transformations, is applied to study the long-term effects of gravity-gradient torque and orbital evolution on the rotational motion of a triaxial, rigid satellite. The center of mass of the satellite is constrained to move in an elliptic orbit about an attracting point mass. The orbit, which has a constant inclination, is constrained to precess and spin with constant rates. The method of general perturbations is used to obtain the Hamiltonian for the nonresonant secular and long-period rotational motion of the satellite to second order in n/omega sub 0, where n is the orbital mean motion of the center of mass and omega sub 0 is a reference value of the magnitude of the satellite's rotational angular velocity. The differential equations derivable from the transformed Hamiltonian are integrable, and the solution for the long-term motion may be expressed in terms of Jacobian elliptic functions and elliptic integrals. Geometrical aspects of the long-term rotational motion are discussed, and a comparison of theoretical results with observations is made.

Cochran, J. E.↗

NASA NTRS1974

Analysis of multiple pulse NMR in solids. II

A systematic method, an extension of the average Hamiltonian formalism, is presented for calculating the effects of pulse errors and imperfections in the multiple pulse nuclear magnetic resonance experiments. Application of this method to account for effects of pulse nonidealities such as phase errors, phase transient effects, pulse size errors, and rf inhomogeneity is found to agree with experimental observation, and the results furnish a basis for understanding the complex couplings between the pulse errors and other interactions such as the dipolar and the chemical shift Hamiltonians.

Rhim, W.-K.↗

NASA NTRS1974

Possible histories of the obliquity of Mercury

Possible evolutionary paths of the spin axis of Mercury are studied theoretically in terms of generalized Cassini's laws using the Hamiltonian approach. Three stable positions (Cassini states) of the spin axis are discerned. The Cassini states in which the spin axis remains fixed in the frame rotating with the orbit precession are extremes in the tide-free Hamiltonian under the condition of constant angular momentum. The stability of the 3/2 spin resonance for Mercury as a function of obliquity is discussed, and it is shown how control of the resonance is passed from one to another term in the averaged potential as the obliquity is increased with a corresponding loss of stability at each transfer. A sampling technique is used to infer the evolution due to tidal interaction. Two tidal models are used - one with a constant value of the specific dissipation function Q and the other with Q inversely proportional to frequency. An approximate analytic solution for precession trajectories very close to a Cassini state is developed.

Peale, S. J.↗

NASA NTRS1975

Four particle exchange in solid He-3

Calculations which demonstrate a physically important four-atom exchange process in bodycentered cubic He crystal and thus an important four-spin term in the exchange Hamiltonian are discussed. A simple, mean-field analysis of this Hamiltonian appears to account for a number of the perplexing properties of bodycentered cubic He crystal. It is suggested that an understanding of other properties may require treatment of the exact four-spin term.

Mcmahan, A. K.↗

NASA NTRS1975

Useful extremum principle for the variational calculation of matrix elements. II

Recent work (Gerjuoy et al., 1974) on variational principles for diagonal bound state matrix elements of arbitrary Hermitian operators is extended. In particular, it is shown that the previously derived minimum principle for the trial auxiliary function appearing in such variational principles can be constructed using a modified Hamiltonian possessing not heretofore recognized positive definite properties. Thus there is at least one alternative to the particular modified Hamiltonian on which the results of Gerjuoy et al. (1974) originally were based.

Gerjuoy, E.↗

NASA NTRS1975

Four-particle exchange in solid He-3

Calculations are reported which suggest that there is a physically important four-atom exchange process in bcc He-3 and an important four-spin term in the exchange Hamiltonian. A simple mean-field analysis of this Hamiltonian appears to account for a number of the perplexing properties of bcc He-3. An understanding of other properties may require treatment of the exact four-spin term.

Mcmahan, A. K.↗

NASA NTRS1976

New treatment of the critical argument in resonance problems

Traditional methods for isolating the critical argument in Hamiltonian mechanics involve the introduction of a special set of action variables which are both complicated in form and arbitrary in nature. The new variables are in general obscure in their interpretation. This paper introduces a new canonical system corresponding to the original one, but with one more degree of freedom, in which all the variables of the original Hamiltonian can be treated on the same footing. It appears to have significant advantages for machine calculations.

Jefferys, W. H.↗

NASA NTRS1978

Critical inclinations in satellite theory

The main problem of satellite theory is described in polar coordinates by a Hamiltonian function. It is proposed to find a solution of the Hamiltonian function with the following properties: (1) the reference orbit is Keplerian; (2) no restriction is imposed on the eccentricity; in particular, it is exempt of singularities - real or apparent - for small eccentricities; and (3) no restriction is imposed on the inclination; in particular, it is exempt of singularities - real or apparent - for small inclinations; also it is valid even in the neighborhood of inclinations at which the perigee is stationary.

Deprit, A.↗

NASA NTRS1979

Acoustic energy in ducts - Further observations

The transmission of acoustic energy in uniform ducts carrying uniform flow is investigated with the purpose of clarifying two points of interest. The two commonly used definitions of acoustic 'energy' flux are shown to be related by a Legendre transformation of the Lagrangian density exactly as in deriving the Hamiltonian density in mechanics. In the acoustic case the total energy density and the Hamiltonian density are not the same which accounts for two different 'energy' fluxes. When the duct has acoustically absorptive walls neither of the two flux expressions gives correct results. A reevaluation of the basis of derivation of the energy density and energy flux provides forms which yield consistent results for soft walled ducts.

Eversman, W.↗

NASA NTRS1982

Stationary state model for normal metal tunnel junction phenomena

A model applicable to normal metal tunnel junctions is presented. This model, referred to herein as the stationary state model, is an extension of the extended basis function theory of Kleinman and Duke (1972). Under the assumption that elastic tunneling is the dominant transport mechanism under static bias, the theory has been extended to include the case where there is an ac component of bias potential, and the fluctuation spectrum has been derived. In this approach structure eigenstates are used as the basis, allowing observables to be evaluated without recourse to the perturbation theory inherent in the transfer Hamiltonian model. Comparison is made to the appropriate results of the first order transfer Hamiltonian model, and it is found that there is close but not always exact agreement to lowest order in the tunneling exponential. The stationary state model should be accurate with large barrier transmission, as it includes all orders of the tunneling exponential. The model as presented here should be applicable to normal metal tunnel junctions, where elastic tunneling is the dominant transport mechanism (under static bias), up to infrared excitation frequencies.

Whiteley, S. R.↗