The Determination of the Conjugate Points for Discontinuous Solutions in the Calculus of Variations

1908 ◽  
Vol 30 (3) ◽  
pp. 209 ◽  
Author(s):  
Oskar Bolza
Keyword(s):  
Calculus Of Variations ◽  
Conjugate Points ◽  
Discontinuous Solutions

The Determination of Maximum Range for an Unpowered Atmospheric Vehicle

1964 ◽  
Vol 68 (638) ◽  
pp. 111-116 ◽  
Author(s):  
D. J. Bell
Keyword(s):  
Calculus Of Variations ◽  
Lagrange Multipliers ◽  
Digital Computer ◽  
Necessary Conditions ◽  
Lagrange Equations ◽  
Initial Portion ◽  
End Conditions ◽  
Maximum Range ◽  
Isothermal Atmosphere

SummaryThe problem of maximising the range of a given unpowered, air-launched vehicle is formed as one of Mayer type in the calculus of variations. Eulers’ necessary conditions for the existence of an extremal are stated together with the natural end conditions. The problem reduces to finding the incidence programme which will give the greatest range.The vehicle is assumed to be an air-to-ground, winged unpowered vehicle flying in an isothermal atmosphere above a flat earth. It is also assumed to be a point mass acted upon by the forces of lift, drag and weight. The acceleration due to gravity is assumed constant.The fundamental constraints of the problem and the Euler-Lagrange equations are programmed for an automatic digital computer. By considering the Lagrange multipliers involved in the problem a method of search is devised based on finding flight paths with maximum range for specified final velocities. It is shown that this method leads to trajectories which are sufficiently close to the “best” trajectory for most practical purposes.It is concluded that such a method is practical and is particularly useful in obtaining the optimum incidence programme during the initial portion of the flight path.

scholarly journals III. On certain ternary alloys. Part V. Determination of various critical curves, and their tie-lines and limiting points

1892 ◽  
Vol 50 (302-307) ◽  
pp. 372-395 ◽  
Keyword(s):  
Graphical Representation ◽  
Conjugate Point ◽  
Critical Curve ◽  
Ternary Mixtures ◽  
Ternary Alloys ◽  
Conjugate Points ◽  
Critical Curves ◽  
Tie Lines ◽  
Tie Line

The triangular method of graphical representation suggested by Sir G. G. Stokes, and described in Part IV (‘Roy. Soc. Proc.,’ vol. 49, p. 174), substantially amounts to the tracing out of a curve (“ critical curve”) which shall express the saturation of the solvent C with a mixture in given variable proportions of the other two constituents, A, B ; the variation being such that any given point on the curve is related to some other point (“ conjugate point ”) in a way given by the consideration that all mixtures of the three constituents, A, B, C, represented by points lying on the line (“ tie-line ”) joining these two conjugate points (“ ideal ” alloys, or mixtures), will separate into two different ternary mixtures corresponding with the two points respectively ; whereas any mixture of the same constituents, repre­sented by a point lying outside the critical curve, will form a “ real ” alloy, or mixture, not separating spontaneously into two different fluids but existing as a stable homogeneous whole.

scholarly journals On the variation of a triple integral

1843 ◽  
Vol 4 ◽  
pp. 42-42
Keyword(s):  
Calculus Of Variations ◽  
General Problem ◽  
The Given

In the calculus of variations, the discovery of which has immor­talized the name of" Lagrange, that illustrious mathematician, by differentiating the function with respect to a new variable which en­ters into it, reduced the general problem of indeterminate maxima and minima to the solution of an equation depending on the variation of the given integral, whether single or multiple, and whose differ­ential coefficient contains any number of variables, or which even de­pends on other integrals. The author investigates, in the present memoir, the case in which the given function is a triple integral; its variation being composed of two distinct parts, namely, a triple inte­gral and another part, the determination of which must be sought from the limits of the triple integral.

scholarly journals The classical theory of calculus of variations for generalized functions

2017 ◽  
Vol 8 (1) ◽  
pp. 779-808 ◽  
Author(s):  
Alexander Lecke ◽  
Lorenzo Luperi Baglini ◽  
Paolo Giordano
Keyword(s):  
Calculus Of Variations ◽  
Classical Theory ◽  
Sufficient Conditions ◽  
Generalized Functions ◽  
Conjugate Points ◽  
Lagrange Equations ◽  
Smooth Functions ◽  
Nonlinear Properties ◽  
Schwartz Distributions ◽  
Necessary And Sufficient

Abstract We present an extension of the classical theory of calculus of variations to generalized functions. The framework is the category of generalized smooth functions, which includes Schwartz distributions, while sharing many nonlinear properties with ordinary smooth functions. We prove full connections between extremals and Euler–Lagrange equations, classical necessary and sufficient conditions to have a minimizer, the necessary Legendre condition, Jacobi’s theorem on conjugate points and Noether’s theorem. We close with an application to low regularity Riemannian geometry.

Sign in / Sign up

Export Citation Format

Share Document