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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1982

Moving frames and prolongation algebras

Differential ideals generated by sets of 2-forms which can be written with constant coefficients in a canonical basis of 1-forms are considered. By setting up a Cartan-Ehresmann connection, in a fiber bundle over a base space in which the 2-forms live, one finds an incomplete Lie algebra of vector fields in the fields in the fibers. Conversely, given this algebra (a prolongation algebra), one can derive the differential ideal. The two constructs are thus dual, and analysis of either derives properties of both. Such systems arise in the classical differential geometry of moving frames. Examples of this are discussed, together with examples arising more recently: the Korteweg-de Vries and Harrison-Ernst systems.

Estabrook, F. B.↗
NASA NTRS1994

A brief survey of constrained mechanics and variational problems in terms of differential forms

There has been considerable interest recently in constrained mechanics and variational problems. This is in part due to applied interests (such as 'non-holonomic mechanics in robotics') and in other part due to the fact that several schools of 'pure' mathematics have found that this classical subject is of importance for what they are trying to do. I have made various attempts at developing these subjects since my Lincoln lab days of the late 1950's. In this Chapter, I will sketch a Unified point of view, using Cartan's approach with differential forms. This has the advantage from the C-O-R viewpoint being developed in this Volume that the extension from 'smooth' to 'generalized' data is very systematic and algebraic. (I will only deal with the 'smooth' point of view in this Chapter; I will develop the 'generalized function' material at a later point.) The material presented briefly here about Variational Calculus and Constrained Mechanics can be found in more detail in my books, 'Differential Geometry and the Calculus of Variations', 'Lie Algebras and Quantum Mechanics', and 'Geometry, Physics and Systems'.

Hermann, Robert↗
NASA NTRS1972

On the algebraic structure of bilinear systems.

It is shown that a particular bilinear model is both quite general and easy to work with. A basic structure theory is developed with the aid of previous results. Some preliminary ideas are discussed together with the system interconnection, the canonical form, questions of controllability, aspects of observability, and equivalent realizations. It is pointed out that in actually determining equivalent realizations for systems and in the classification of systems, the results available in the study of Lie algebras are of fundamental importance.

Brockett, R. W.↗
DOE OSTI2026

SCF Framework, HF Stability, and RPA Correlation for Jordan–Wigner-Transformed Spin Hamiltonians on Arbitrary Coupling Topologies

Mapping spins to fermions via the Jordan–Wigner (JW) transformation can render mean-field (Hartree–Fock, HF) descriptions effective for strongly correlated spin systems. As established in recent work, the application of such approaches is not limited by the nonlocal structure of JW strings or by site ordering because string operators can be absorbed into Thouless rotations of a Slater determinant, and the variational optimization of a unitary Lie-algebraic similarity transformation removes any ordering dependence. Leveraging these ideas, we develop a self-consistent field (SCF) scheme that expresses the mean-field energy as a functional of the single-particle density matrix, providing an alternative to gradient-based optimization of Thouless parameters. We derive the analytical orbital Hessian to diagnose HF stability and compute the ground-state correlation energy through the random-phase approximation (RPA). Benchmark results for the XXZ and J 1 –J 2 model on one- and two-dimensional lattices demonstrate that RPA significantly improves mean-field accuracy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Fractonic coset construction for spontaneously broken translations

We study the homogeneous breaking of spatial translation symmetry concomitantly with the spontaneous breaking of other internal and spacetime symmetries, including dilations. We use the symmetry-breaking pattern as the only input to derive, via the coset construction, general effective field theories for the symmetry-originated modes associated with Goldstone’s theorem, namely the Nambu-Goldstone candidates. Through explicit computations, we show that integrating out the explicit massive Nambu-Goldstone candidates or imposing symmetric constraints, namely the inverse Higgs constraints, to express massive modes in terms of the massless ones leads to physically distinct effective field theories. This sensitivity to the chosen method can be traced back to the homogeneous breaking of translations, the homogeneous aspect of the breaking induces a mixing between internal and spacetime symmetries at the level of the Lie algebra. This, in turn, leads to subtle discussions about the inverse Higgs constraints, in particular that they lead to a loss of generality in our specific examples. The derived general effective field theories also give rise to a broad class of theories exhibiting emergent enhanced shift symmetries, which constrain the mobility of the modes. The latter are referred to as fractonic modes. Published by the American Physical Society 2025

Chavda, Ameya (ORCID:0000000211731605)↗
DOE OSTI2026

Quantum simulation of massive Thirring and Gross--Neveu models for arbitrary number of flavors

The study of fermionic quantum field theories is an important problem for realizing the standard model of particle physics on a quantum computer. As a step towards this goal, we consider the massive Thirring and Gross--Neveu models with arbitrary number of fermion flavors, $N_f$, discretized on a spatial one-dimensional lattice of size $L$ in the Hamiltonian formulation. We compute the gate complexity using the higher-order product formula and using block-encoding/qubitization and quantum singular value transformations in the limit of large $N_f$ and $L$. We also prepare the ground states of both models with excellent fidelity for system sizes up to 20 qubits with $N_f = 1,2,3,4$ using the adaptive-variational quantum imaginary time algorithm. In addition, we also classify the dynamical Lie algebras of these relativistic fermionic models and show that they belong to the same isomorphism class. Our work is a concrete step towards the quantum simulation of real-time dynamics of large $N_f$ fermionic quantum field theories models relevant for chiral symmetry breaking, understanding dimensional transmutation, and exploring the conformal window of field theories on near-term and early fault-tolerant quantum computers.

FOS: Physical sciences↗
NASA NTRS1972

System theory on group manifolds and coset spaces.

The purpose of this paper is to study questions regarding controllability, observability, and realization theory for a particular class of systems for which the state space is a differentiable manifold which is simultaneously a group or, more generally, a coset space. We show that it is possible to give rather explicit expressions for the reachable set and the set of indistinguishable states in the case of autonomous systems. We also establish a type of state space isomorphism theorem. Our objective is to reduce all questions about the system to questions about Lie algebras generated from the coefficient matrices entering in the description of the system and in that way arrive at conditions which are easily visualized and tested.

Brockett, R. W.↗
NASA NTRS1972

Control systems on Lie groups.

The controllability properties of systems which are described by an evolution equation in a Lie group are studied. The revelant Lie algebras induced by a right invariant system are singled out, and the basic properties of attainable sets are derived. The homogeneous case and the general case are studied, and results are interpreted in terms of controllability. Five examples are given.

Jurdjevic, V.↗
NASA NTRS1974

Power conversion in electrical networks

Aspects of dc to dc conversion were studied in terms of a class of switching voltage regulators from a stability viewpoint. Background concepts of nonlinear system theory were considered, including the problem of obtaining suitable realizations for a class of positive operators. It is shown that the state evolution equations for a power conversion network are in general of bilinear form, and that the theory of lie groups and lie algebras is useful in analyzing such systems. The feedback stabilization of a class of bilinear systems whose state space is a manifold is also discussed.

Wood, J. R.↗
NASA NTRS1974

Switched electrical networks and bilinear equations

State equations arising in the description of power processing systems are described. The role played by Lie groups and Lie algebras in characterizing the inherent dynamical features of these systems is outlined, and network examples are presented for illustration.

Wood, J. R.↗
NASA NTRS1974

Switched electrical networks and bilinear equations

An investigation is conducted concerning the state equations which arise in the description of power processing systems. Attention is given to the role played by Lie groups and Lie algebras in the characterization of the dynamical features of the systems. The bilinear equations used for the representation of the network characteristics are discussed along with the nature of the solutions for the equations. The application of the described approaches is illustrated with the aid of a number of network examples.

Wood, J. R.↗
NASA NTRS1975

Controllability in nonlinear systems

An explicit expression for the reachable set is obtained for a class of nonlinear systems. This class is described by a chain condition on the Lie algebra of vector fields associated with each nonlinear system. These ideas are used to obtain a generalization of a controllability result for linear systems in the case where multiplicative controls are present.

Hirschorn, R. M.↗
NASA NTRS1980

A remarkable operator version of the Navier-Stokes equation

The Navier-Stokes equation is transformed into a closed nonlinear equation for the convected curl operator. It is shown how Lie algebraic methods can be employed to solve the obtained double-commutator equation. The proposed operator equation provides a way to the further development of the Navier-Stokes fluid flow theory.

Rosen, G.↗
NASA NTRS1979

Classification and equivalence in estimation theory

A method is proposed for classifying estimation problems based on the Lie algebra generated by the operators which appear in the conditional density equation. A natural class of automorphisms of this algebra is examined and a systematic method of generating equivalent problems is developed. Finally, a new class of nonlinear filtering problems with essentially nonlinear filtering equations are presented.

Brockett, R. W.↗
NASA NTRS1980

Lagrangian methods in nonlinear plasma wave interaction

Analysis of nonlinear plasma wave interactions is usually very complicated, and simplifying mathematical approaches are highly desirable. The application of averaged-Lagrangian methods offers a considerable reduction in effort, with improved insight into synchronism and conservation (Manley-Rowe) relations. This chapter indicates how suitable Lagrangian densities have been defined, expanded, and manipulated to describe nonlinear wave-wave and wave-particle interactions in the microscopic, macroscopic and cold plasma models. Recently, further simplifications have been introduced by the use of techniques derived from Lie algebra. These and likely future developments are reviewed briefly.

Crawford, F. W.↗
NASA NTRS1982

Global differential geometry: An introduction for control engineers

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

Doolin, B. F.↗
NASA NTRS1990

Trees, bialgebras and intrinsic numerical algorithms

Preliminary work about intrinsic numerical integrators evolving on groups is described. Fix a finite dimensional Lie group G; let g denote its Lie algebra, and let Y(sub 1),...,Y(sub N) denote a basis of g. A class of numerical algorithms is presented that approximate solutions to differential equations evolving on G of the form: dot-x(t) = F(x(t)), x(0) = p is an element of G. The algorithms depend upon constants c(sub i) and c(sub ij), for i = 1,...,k and j is less than i. The algorithms have the property that if the algorithm starts on the group, then it remains on the group. In addition, they also have the property that if G is the abelian group R(N), then the algorithm becomes the classical Runge-Kutta algorithm. The Cayley algebra generated by labeled, ordered trees is used to generate the equations that the coefficients c(sub i) and c(sub ij) must satisfy in order for the algorithm to yield an rth order numerical integrator and to analyze the resulting algorithms.

Crouch, Peter↗
NASA NTRS1993

Point form relativistic quantum mechanics and relativistic SU(6)

The point form is used as a framework for formulating a relativistic quantum mechanics, with the mass operator carrying the interactions of underlying constituents. A symplectic Lie algebra of mass operators is introduced from which a relativistic harmonic oscillator mass operator is formed. Mass splittings within the degenerate harmonic oscillator levels arise from relativistically invariant spin-spin, spin-orbit, and tensor mass operators. Internal flavor (and color) symmetries are introduced which make it possible to formulate a relativistic SU(6) model of baryons (and mesons). Careful attention is paid to the permutation symmetry properties of the hadronic wave functions, which are written as polynomials in Bargmann spaces.

Klink, W. H.↗