Modeling and adjoints for continuous systems

Approximate solution of the differential equations of state of continuous systems by various numerical integration schemes is standard practice in trajectory optimization and control work; the resulting truncation error represents the main error in many applications. Suppression of the deleterious effects of this error is of increasing interest as double-precision arithmetic becomes routinely available for roundoff error reduction, especially when high overall accuracy is needed. In numerical optimization of trajectories, the accuracy of partial derivatives is important in a special sense: compatibility of a function and its derivatives. That is, if the partial derivatives of the terminal state with respect to trajectory parameters are accurate representations of the partial derivatives of the terminal state calculated through the integration model, adverse effects on convergence of successive approximation iterations can be avoided. From previous numerical experiments, this is known to be particularly important when conjugate-direction methods are used, as they require unusually accurate first partial derivatives to realize the quadratic convergence theoretically attainable.When the mathematical model of the system consists of differential equations, it is theoretically correct to use differential equations for the adjoint variables (influence functions) as well. However, the actual numerical solution of both state and adjoint variables is obtained by using finite-difference approximations. Truncation errors in the state are inevitable; but, by suitable construction of the adjoint difference equation, it is possible to obtain the compatibility just discussed. The procedure is to recognize that, for numerical purposes, the system model consists of difference equations and directly derive compatible adjoint difference equations. The adjoint variables are then the correct influence functions for the numerically computed state variables, rather than approximate influence functions for the theoretically continuous state variables. The construction of the adjoint system for difference-equation models is straightforward. For example, if the typical integration routine involves fourth-order differences, the adjoint system also involves fourth-order differences. However, the intervals and coefficients for the adjoint system differ from those of the dynamic system. Thus, the compatible adjoint system uses difference equations of the same order in a different, although quite connected, integration routine.This paper was presented at the Second International Conference on Computing Methods in Optimization Problems, San Remo, Italy, 1968. The research reported in this paper was carried out under Contract No. NAS 9-7805 with the NASA Manned Spacecraft Center, Houston, Texas.     

A note on isoenergetic stability

In this note we consider certain two-degree-of-freedom Hamiltonian systems which may be regarded as perturbations of integrable systems governed by a real parameter ε. We wish to study the stability, at fixed energy, of certain periodic solutions. Two constants are defined, computable in terms of the original Hamiltonian function and the energy. The main theorem then states that if these constants are not zero, the periodic solutions are isoenergetically stable for sufficiently small ε. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser. By way of illustration, we apply the theorem to a mechanical system consisting of coupled non-linear oscillators. The periodic solutions are the “normal modes” and ε governs the non-linearity of the system. One obtains stability criteria for arbitrary energies and small ε, or, alternatively, for arbitrary ε and small energies.     

Massenspektrometrische Untersuchungen von Terpenen

Zusammenfassung Die Massenspektren von 9 Monoterpen-aldehyden und -ketonen (Isomenthon, cis- und trans-Caranon-(3), Isopiperitenon, Carvon, Eucarvon, Verbenon, Perilla-aldehyd und Myrtenal) wurden ausgewertet und für die Zwecke der qualitativen Terpenanalyse durch Zahlenwerte der Intensitäten der jeweils 10 häufigsten Ionen charakterisiert. Sie unterscheiden sich alle deutlich voneinander; selbst die stereoisomeren Caranone-(3) lassen sich aufgrund von Intensitätsverhältnissen charakteristischer Fragment-Ionen eindeutig identifizieren. Aus der Untersuchung der metastabilen Ionen ergaben sich Hinweise auf den Mechanismus der lonenfragmentierung. Die Verwendung von Ionisierungs- und Auftrittspotentialdaten für die Strukturaufklärung wird diskutiert.

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Summary The mass spectra of 9 monoterpene-aldehydes and -ketones (isomenthone, eis- and trans-caranone-(3), isopiperitenone, carvone, eucarvone, verbenone, perillaldehyde and myrtenal) are evaluated and for purposes of qualitative terpene analysis each is characterized by numerical values of the intensities of the 10 most abundant ions. All mass spectra exhibit significant differences: even the stereoisomeric caranones-(3) can be identified unambiguously by the abundance ratios of characteristic fragment ions. Features of the ion fragmentation mechanisms could be derived from metastable ion analysis. The use of ionization- and appearancepotential data for structure elucidation is discussed.

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Herrn Prof. Dr. G. O. Schenck danken wir für die wohlwollende Förderung dieser Arbeit und für einige wichtige Hinweise. Den Herren Dr. G. Schaden und Dr. G. Steffan danken wir für anregende Diskussionen.     

The problem of causality in an indeterministic science

A differentiation between the principles of causality and strict determinism is suggested, the principle of strict determinism being considered to be incompatible with current scientific theory, while the principle of causality is supported by all contemporary scientific knowledge. In this paper the importance of the principle of causality in indeterministic causal theories is discussed, non-relativistic quantum mechanics being considered in detail. Indeterministic causality and its relation to relativistic quantum theory is also discussed.     

The geometrical and vibrational properties of the Rh(111) surface

Low-energy electron diffraction (LEED) data have been used to characterize the clean Rh(111) surface. The surface geometry, the degree of surface relaxation, and the Debye temperature have been determined. In the Debye temperature measurement, specular LEED beam intensities were monitored as a function of temperature over a range of electron energies from approximately 30 to 1000 eV. It was found that the bulk Debye temperature is 380 ± 23 K, and the normal component of the Debye temperature at the lowest electron energy used is 197 ± 12 K. The Rh(111) surface relaxation has been determined both by a convolution-transform analysis and by dynamical calculations. Within experimental error, neither expansion nor contraction of the topmost layer has been detected. The results of the convolution-transform analysis of specular beams at two angles of incidence and of a nonspecular beam at normal incidence suggest an expansion of the topmost layer of 3 ± 5% of the bulk layer spacing. In agreement with this, comparisons between the results of the dynamical calculation and experimental data for five nonspecular beams at normal incidence suggest that the surface layer relaxes by 0 ± 5%. In addition, the dynamical calculations indicate that the topmost layer maintains an fcc structure.     

On the use of incomplete constitutive information in thermoelasticity

The extent to which non-linear thermoelastic constitutive data may be determined by controllable states is delineated, and thermomechanical states that may be analyzed completely with such incomplete data are catalogued. These include non-homogeneous finite deformations coupled with quite general temperature fields in plane, cylindrical and spherical geometries. Two problems involving the states are worked out as examples.This research was supported by the U.S. National Science Foundation Grant GK-37367 to The University of Texas at Austin.