Search Results - (Author, Cooperation:C. D. Bailey)

2014-02-18

American Association for the Advancement of Science (AAAS)

0036-8075

1095-9203

Biology

Chemistry and Pharmacology

Computer Science

Medicine

Natural Sciences in General

Physics

Amino Acid Sequence ; Arabidopsis/anatomy & histology/genetics ; Brassicaceae/*anatomy & histology/*genetics ; Chromosome Mapping ; *Evolution, Molecular ; Gene Duplication ; *Gene Expression Regulation, Plant ; *Genes, Homeobox ; Genetic Complementation Test ; Molecular Sequence Data ; Plant Leaves/*anatomy & histology/*genetics

1619-6937

Springer Online Journal Archives 1860-2000

Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics

Physics

Zusammenfassung Die Galerkinsche und die Hamilton-Ritzsche Formulierung: Ein Vergleich. Der Zweck dieser Arbeit ist zu zeigen, warum die Lösungen für kontinuierliche Systeme, seien sie konservativ oder nicht, stationär oder instationär, nicht aus der Galerkinschen Formulierung mit Hilfe gewöhnlicher Potenzreihenentwicklungen so einfach, wie es in unseren Veröffentlichungen gezeigt wurde, erhalten werden können. Das Hamiltonsche Gesetz und das Hamiltonsche Prinzip werden besprochen. Die Galerkinsche Formulierung wird auf Differentialgleichungen angewendet. Der Ursprung der Differentialgleichungen ist dabei immateriell. Das Ergebnis wird mit dem durch Anwendung der Ritzschen Methode auf das Hamiltonsche Gesetz, also aus der Hamilton-Ritz-Formulierung gefundenem, verglichen. Das Ergebnis zeigt deutlich, warum unsere Lösungen für Kontinua, ausgenommen für einfach beschreibbare Spezialfälle, nicht durch eine strenge, direkte Anwendung der Galerkinschen Methode erhalten werden konnten und können.

Summary The Galerkin Formulation and the Hamilton-Ritz Formulation: A Comparison. It is the purpose of this paper to show why the solutions to continuous systems, whether conservative or nonconservative, stationary or nonstationary, cannot be achieved from the Galerkin formulation by means of the simple power series with the simplicity now demonstrated in our published papers. Hamilton's law and Hamilton's principle are discussed. The Galerkin formulation is applied to the differential equation. The source of the differential equation is immaterial. The result is compared to the Ritz method applied to Hamilton's law which is called the Hamilton-Ritz formulation. The result clearly demonstrates why our direct analytical solutions for continuua, except for easily identifiable special cases, have not been and cannot be obtained through a correct, direct application of the Galerkin method.

Electronic Resource

1619-6937

Springer Online Journal Archives 1860-2000

Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics

Physics

Summary In the recent literature of the Calculus of variations, mathematical proofs have been presented for what the writers claim to be a more precise statement of Hamilton's Principle for conservative systems. Nothing is said about Hamilton's Principle for nonconservative systems. According to these writers, the action integral, the variation of which is Hamilton's Principle for conservative systems, is a minimum for discrete systems for small time intervals only and is never a minimum for continuous systems. The proof of this more precise statement is based in the sufficiency theorems of the Calculus of Variations. In this paper, two contradictions to the statement are demonstrated — one for a discrete system and one for a continuous system.

Electronic Resource