This content is not included in
your SAE MOBILUS subscription, or you are not logged in.
Solution of the Airplane Minimum Time Problem by Huygens' Principle
Annotation ability available
Sector:
Language:
English
Abstract
The relationship between Huygen's principle in geometrical optics and the calculus of variations is examined. It is found that the calculus of variations is a straightforward generalization of the theory of geometrical optics. As a consequence, a new and simplified approach to the calculus of variations is obtained. This approach reduces problems in the calculus of variations to problems in ordinary maxima and minima in the calculus. The approach rests heavily on the early work of C. Huygens, 1690, and John Bernoulli, 1718, and its extension by W. Hamilton and C. Caratheodory. New numerical techniques, suggested by this approach are examined and compared and two of these are applied to the airplane minimum time problem.
Authors
Topic
Citation
Heermann, H., Kretsinger, P., and McCorkle, B., "Solution of the Airplane Minimum Time Problem by Huygens' Principle," SAE Technical Paper 650784, 1965, https://doi.org/10.4271/650784.Also In
References
- Heermann H. Kretsinger P. “The Minimum Time Problem,” Journal of the Astronautical Sciences X 4 1964 93 107
- Huygens C. “Traite de la Luminere” 1690 Chicago University of Chicago Press 1945
- Caratheodory C. “Uber diskontinuierliche Losungen in der Variationsrechnung,” Dissertation Gottingen 1904
- Caratheodory C. “The Beginning of Research in the Calculus of Variations,” OSIRIS III 1938 224 240
- Woodhouse R. “A Treatise on Isoperimetrical Problems and the Calculus of Variations,” 1810 Chelsea Publishing Co. 1965
- Kalman R. “The Theory of Optimal Control and the Calculus of Variations,” Chap. 16 in Mathematical Optimization Techniques Berkeley University of California Press 1963
- Bellman R. Dreyfus S. “Applied Dynamic Programming,” Princeton, N. J. Princeton University Press 1962
- Bellman R. “Adaptive Control Processes,” Princeton, N. J. Princeton University Press 1961
- Pontryagin L. Boltyanskii V. Gamkrelidze R. Misbchenko E. “The Mathematical Theory of Optimal Processes,” New York Interscience 1962
- Halkin H. “The Principle of Optimal Evolution,” Nonlinear Differential Equations and Nonlinear Mechanics New York Academic Press 1963 284 302
- Roxin E. “A Geometric Interpretation of Pontryagin's Maximum Principle.” Nonlinear Differential Equations and Nonlinear Mechanics New York Academic Press 1963 303 332
- Paiewonsky B. “The Optimal Control of Linear Systems with Bounded Controls,” Nonlinear Differential Equations and Nonlinear Mechanics New York Academic Press 1963 333 365
- Dreyfus S. E. “Dynamic Programming and the Calculus of Variations,” Rand Report 1464 1958
- Courant R. “Methods of Mathematical Physics,” II New York Interscience 1953
- Bliss G. A. “Lectures on the Calculus of Variations,” Chicago, Ill. Univ. of Chicago Press 1945
- Kline M. Kay I. “Electromagnetic Theory and Geometrical Optics.” New York Interscience 1965
- Born M. “Optik” Berlin Julius Springer 1933
- Caratheodory C. “Geometrische Optik,” Berlin Julius Springer 1937
- Bliss G. A. “Calculus of Variations,” Chicago, Ill. Open Court Publishing Co. 1935
- Bell E. T. “The Development of Mathematics,” New York McGraw-Hill Co. 1940
- Smith D. E. “A Source Book in Mathematics,” New York McGraw-Hill Co. 1929
- Todhunter I. “A History of the Calculus of Variations During the Nineteenth Century,” New York Chelsea Publishing Co.
- Synge J. L. “The Life and Early Work of Sir William Rowan Hamilton,” in a collection of papers in memory of Sir William Rowan Hamilton, Scripta Mathematica 1945
- Bellman R. “Dynamic Programming and a New Formalism in the Calculus of Variations,” Proceedings of the National Academy of Sciences 40 1954 231 235
- Hestenes M. R. “A General Problem in the Calculus of Variations with Applications to Paths of Least Time,” Rand Report RM-100 1950
- Breakwell J. V. “The Optimization of Trajectories,” Journal of the Society of Industrial and Applied Mathematics 7 1959
- Miele A. “General Solutions of Optimum Problems in Nonstationary Flight,” NACA TM 1388 1955
- Caratheodory C. “Gesammelte Mathematische Schriften,” BAND I-V Munchen, C. H. Beck'sche Verlags-buchhandlung 1954
- Rutowski E. “Energy Approach to the General Aircraft Performance Problem,” Journal of the Aeronautical Sciences 21 3 1954