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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
NASA NTRS1998

Squeezed states as representations of symplectic groups

It is shown that the SU(1,1)-like and SU(2)-like two-photon coherent states can be combined to form a O(3,2)-like two-photon state. Since the O(3,2) group has many subgroups, there are also many new interesting new coherent and squeezed two-photon states. Among them is the two-photon sheared state whose symmetry property is like that for the two-dimensional Euclidean group. There are now two-phonon coherent states which may exhibit symmetries not yet observed for photons, including sheared states. Let us note that both SU(1,1) and S(3,2) are isomorphic to the symplectic groups Sp(2) and Sp(4) respectively, and that symplectic transformations consist of rotations and squeeze transformations.

Han, D.↗
NASA NTRS2012

A Survey of Symplectic and Collocation Integration Methods for Orbit Propagation

Demands on numerical integration algorithms for astrodynamics applications continue to increase. Common methods, like explicit Runge-Kutta, meet the orbit propagation needs of most scenarios, but more specialized scenarios require new techniques to meet both computational efficiency and accuracy needs. This paper provides an extensive survey on the application of symplectic and collocation methods to astrodynamics. Both of these methods benefit from relatively recent theoretical developments, which improve their applicability to artificial satellite orbit propagation. This paper also details their implementation, with several tests demonstrating their advantages and disadvantages.

integration↗
NASA NTRS1982

The Riccati equation, imprimitive actions and symplectic forms

This paper looks at the structure of the solution of a matrix Riccati differential equation under a predefined group of transformations. The group of transformations used is an expanded form of the feedback group. It is shown that this group of transformations is a subgroup of the symplectic group. The orbits of the Riccati differential equation under the action of this group are studied and it is seen how these techniques apply to a decentralized optimal control problem.

Garzia, M. R.↗
NASA NTRS1991

Symplectic maps for the n-body problem

The present study generalizes the mapping method of Wisdom (1982) to encompass all gravitational n-body problems with a dominant central mass. The rationale for the generalized mapping method is discussed as well as details for the mapping for the n-body problem. Some refinements of the method are considered, and the relationship of the mapping method to other symplectic integration methods is shown. The method is used to compute the evolution of the outer planets for a billion years. The resulting evolution is compared to the 845 million year evolution of the outer planets performed on the Digital Orerry using standard numerical integration techniques. This calculation provides independent numerical confirmation of the result of Sussman and Wisdom (1988) that the motion of the planet Pluto is chaotic.

Wisdom, Jack↗
NASA NTRS1992

Symplectic maps for the n-body problem - Stability analysis

The stability of new symplectic n-body maps is examined from the point of view of nonlinear dynamics. The resonances responsible for the principal artifacts are identified. These are resonances between the stepsize and the difference of mean motions between pairs of planets. For larger stepsizes resonant perturbations are evident in the variation of the energy of the system corresponding to these stepsize resonances. It is shown that the principal instability of the method can be predicted and corresponds to the overlap of the stepsize resonances. It is noted that the analysis suggests that other artifacts will occur. For example, the overlap of a stepsize resonance with a resonance of the actual system may also give a region of chaotic behavior that is an artifact. It is pointed out that the fact that the principal artifacts corresponds to a particular set of stepsize resonances suggests that it may be possible to perturbatively remove the effect when the stepsize resonances are nonoverlapping.

Wisdom, Jack↗
NASA NTRS1995

Galactic oscillator symmetry

Riemann ellipsoids model rotating galaxies when the galactic velocity field is a linear function of the Cartesian coordinates of the galactic masses. In nuclear physics, the kinetic energy in the linear velocity field approximation is known as the collective kinetic energy. But, the linear approximation neglects intrinsic degrees of freedom associated with nonlinear velocity fields. To remove this limitation, the theory of symplectic dynamical symmetry is developed for classical systems. A classical phase space for a self-gravitating symplectic system is a co-adjoint orbit of the noncompact group SP(3,R). The degenerate co-adjoint orbit is the 12 dimensional homogeneous space Sp(3,R)/U(3), where the maximal compact subgroup U(3) is the symmetry group of the harmonic oscillator. The Hamiltonian equations of motion on each orbit form a Lax system X = (X,F), where X and F are elements of the symplectic Lie algebra. The elements of the matrix X are the generators of the symplectic Lie algebra, viz., the one-body collective quadratic functions of the positions and momenta of the galactic masses. The matrix F is composed from the self-gravitating potential energy, the angular velocity, and the hydostatic pressure. Solutions to the hamiltonian dynamical system on Sp(3,R)/U(3) are given by symplectic isospectral deformations. The Casimirs of Sp(3,R), equal to the traces of powers of X, are conserved quantities.

Rosensteel, George↗
NASA NTRS1976

A note on the solution of the variational equations of a class of dynamical systems

Some properties are derived for the solutions of the variational equations of a class of dynamical systems. It is shown that under rather general conditions, the matrix of the linearized Lagrangian equations of motion have an important property for which the word 'skew-symplectic' has been introduced. It is also shown that the fundamental matrix of solutions is 'symplectic', the word symplectic being used here in a more general sense than in the classical literature. Two consequences of the symplectic property are that the fundamental matrix is easily invertible and that the eigenvalues appear in reciprocal pairs. The effect of coordinate transformations is also analyzed; in particular, the change from Lagrangian to canonical systems.

Broucke, R.↗
NASA NTRS2002

Solar System Dynamics

In these 18 years, the research has touched every major dynamical problem in the solar system, including: the effect of chaotic zones on the distribution of asteroids, the delivery of meteorites along chaotic pathways, the chaotic motion of Pluto, the chaotic motion of the outer planets and that of the whole solar system, the delivery of short period comets from the Kuiper belt, the tidal evolution of the Uranian arid Galilean satellites, the chaotic tumbling of Hyperion and other irregular satellites, the large chaotic variations of the obliquity of Mars, the evolution of the Earth-Moon system, and the resonant core- mantle dynamics of Earth and Venus. It has introduced new analytical and numerical tools that are in widespread use. Today, nearly every long-term integration of our solar system, its subsystems, and other solar systems uses algorithms that was invented. This research has all been primarily Supported by this sequence of PGG NASA grants. During this period published major investigations of tidal evolution of the Earth-Moon system and of the passage of the Earth and Venus through non-linear core-mantle resonances were completed. It has published a major innovation in symplectic algorithms: the symplectic corrector. A paper was completed on non-perturbative hydrostatic equilibrium.

Wisdom, Jack↗
NASA NTRS1999

Dynamical Chaos in the Wisdom-Holman Integrator: Origins and Solutions

We examine the nonlinear stability of the Wisdom-Holman (WH) symplectic mapping applied to the integration of perturbed, highly eccentric (e-0.9) two-body orbits. We find that the method is unstable and introduces artificial chaos into the computed trajectories for this class of problems, unless the step size chosen 1s small enough that PeriaPse is always resolved, in which case the method is generically stable. This 'radial orbit instability' persists even for weakly perturbed systems. Using the Stark problem as a fiducial test case, we investigate the dynamical origin of this instability and argue that the numerical chaos results from the overlap of step-size resonances; interestingly, for the Stark-problem many of these resonances appear to be absolutely stable. We similarly examine the robustness of several alternative integration methods: a time-regularized version of the WH mapping suggested by Mikkola; the potential-splitting (PS) method of Duncan, Levison, Lee; and two original methods incorporating approximations based on Stark motion instead of Keplerian motion. The two fixed point problem and a related, more general problem are used to conduct a comparative test of the various methods for several types of motion. Among the algorithms tested, the time-transformed WH mapping is clearly the most efficient and stable method of integrating eccentric, nearly Keplerian orbits in the absence of close encounters. For test particles subject to both high eccentricities and very close encounters, we find an enhanced version of the PS method-incorporating time regularization, force-center switching, and an improved kernel function-to be both economical and highly versatile. We conclude that Stark-based methods are of marginal utility in N-body type integrations. Additional implications for the symplectic integration of N-body systems are discussed.

Rauch, Kevin P.↗
NASA NTRS2014

Coordinated Mineralogical and Isotopic Analysis of a Cosmic Symplectite Identified in a Stardust Terminal Particle

Comet Wild-2 samples returned by the Stardust spacecraft contain a chemically diverse mixture of material, underscoring the complex nature of comets. Studies of entire Stardust aerogel tracks afford the opportunity to examine the fine-grained particle fragments distributed along the length of the track as well as the terminal particles. Previous TEM characterization of a terminal particle (TP) in track #147 revealed a symplectically intergrown iron sulfide and oxide assemblage. Mineralogically similar assemblages, known as cosmic symplectites (COS, formerly termed "new-PCP"), have only been identified in the primitive carbonaceous chondrite Acfer 094. Meteoritic COS have isotopically heavy O compositions (delta (sup 17), O-18 = 180per mille) that point to interactions with early solar system primordial water. In this study we report mineralogical and O isotopic measurements of the Wild-2 COS assemblage. Experimental: Track #147 is a "bulbous"-type track (4600 microns long) containing 7 terminal particles. The TPs were removed from the track, embedded in epoxy, and ultramicrotomed. A JEOL 2500SE 200 keV field-emission scanning-transmission electron microscope was used to obtain quantitative elemental maps and detailed mineralogical characterization. Following TEM analysis, two thin sections of TP4 (12 microns) were analyzed for O isotopes by raster ion imaging with the JSC NanoSIMS 50L. All three O isotopes were measured simultaneously using electron multipliers. San Carlos olivine grains were used as isotopic standards. Results and Discussion: The COS in the Wild-2 track #147 TP4 sample consists of symplectically intergrown pentlandite and nanocrystalline maghemite which coexists with high-Ca pyroxene with Na and Cr (kosmochlor component). This kosmochlor component could have a nebular origin and be precursors to type II chondrules in ordinary chondrites. Yet pentlandite is not a stable phase in the nebula. The COS in Acfer 094 also consists of pentlandite, but contains magnetite [4] rather than the more oxidized maghemite observed in the Wild-2 COS. The Acfer 094 COS display heavy O isotopic compositions that are the result of sulfidization and oxidation of Fe, Ni-metal grains and sulfides by O-17- and O-18-rich water in the solar nebula or possibly on the parent body. The O isotopic composition of the Wild-2 COS, however, is indistinguishable from terrestrial, indicating it was not altered by the same primordial aqueous reservoir as the Acfer 094 COS. The alteration could have occurred on the parent body by isotopically equilibrated ice. The mineralogy and petrography of Wild-2 samples suggests an incomplete or nascent hydration process. In future work we will analyze S isotopes in the track #147 TP4 COS and search for additional COS in Stardust samples.

Nguyen, A. N.↗
NASA NTRS1976

Microcracks, micropores, and their petrologic interpretation for 72415 and 15418

Lunar samples 72415 and 15418 have complex microstructures that indicate a series of fracturing and healing events. Both samples contain relatively few open microcracks but many sealed and healed microcracks. Dunite 72415 contains abundant healed cracks that formed tectonically, symplectic intergrowths spatially and probably genetically related to microcracks, and a cataclastic matrix that has been extensively sintered. Metamorphosed breccia 15418 contains many post-metamorphic healed cracks, large shock induced cracks that have been sealed with glass, and a few younger, thin, open shock induced cracks.

Richter, D.↗
NASA NTRS1983

Riccati group invariants of linear hamiltonian systems

The action of the Riccati group on the Riccati differential equation is associated with the action of a subgroup of the symplectic group on a set of hamiltonian matrices. Within this framework various sets of canonical forms are developed for the matrix coefficients of the Riccati differential equation. The canonical forms presented are valid for arbitrary Kronecker indices, and it is shown that the Kronecker indices are invariants for this group action. These canonical forms are useful for studying problems arising in the areas of optimal decentralized control and the spectral theory of optimal control problems.

Garzia, M. R.↗
NASA NTRS1982

Linear Hamiltonian systems - The Riccati group and its invariants

The action of the Riccati group on the Riccati differential equation is associated with the action of a subgroup of the symplectic group on a set of Hamiltonian matrices. Within this framework canonical forms are developed for the matrix coefficients of the Riccati differential equation.

Garzia, M. R.↗
NASA NTRS1986

Refractory minerals in interplanetary dust

A newly studied interplanetary dust particle contains a unique set of minerals that closely resembles assemblages in the refractory, calcium- and aluminum-rich inclusions in carbonaceous chondrite meteorites. The set of minerals includes diopside, magnesium-aluminum spinel, anorthite, perovskite, and fassaite. Only fassaite has previously been identified in interplanetary dust particles. Diopside and spinel occur in complex symplectic intergrowths that may have formed by a reaction between condensed melilite and the solar nebula gas. The particle represents a new link between interplanetary dust particles and carbonaceous chondrites; however, the compositions of its two most abundant refractory phases, diopside and spinel, differ in detail from corresponding minerals in calcium- and aluminum-rich inclusions.

Christoffersen, Roy↗
NASA NTRS1991

Mapping the stability field of Jupiter Trojans

Jupiter Trojans are a remnant of outer solar system planetesimals captured into stable or quasistable libration about the 1:1 resonance with the mean motion of Jupiter. The observed swarms of Trojans may provide insight into the original mass of condensed solids in the zone from which the Jovian planets accumulated, provided that the mechanisms of capture can be understood. As the first step toward this understanding, the stability field of Trojans were mapped in the coordinate proper eccentricity, e(sub p), and libration amplitude, D. To accomplish this mapping, the orbits of 100 particles with e(sub p) in the range of 0 to 0.8 and D in the range 0 to 140 deg were numerically integrated. Orbits of the Sun, the four Jovian planets, and the massless particles were integrated as a full N-body system, in a barycentric frame using fourth order symplectic scheme.

Levison, H. F.↗