Analogues of the Jacobi Condition for the Problem of Mayer in the Calculus of Variations

1934 ◽  
Vol 35 (4) ◽  
pp. 836 ◽  
Author(s):  
William T. Reid
Keyword(s):  
Calculus Of Variations ◽  
Jacobi Condition

Sufficiency and the Jacobi Condition in the Calculus of Variations

1986 ◽  
Vol 38 (5) ◽  
pp. 1199-1209 ◽  
Author(s):  
Frank H. Clarke ◽  
Vera Zeidan
Keyword(s):  
Calculus Of Variations ◽  
Basic Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Condition ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Differentiable Mappings

Besides stating the problem and the results, we shall give in this section a brief overview of the classical necessary and sufficient conditions in the calculus of variations, in order to clearly situate the contribution of this article.1.1 The problem. We are given an interval [a, b], two points xa, xb in Rn, and a function L (the Lagrangian) mapping [a, b] × Rn × Rn to R. The basic problem in the calculus of variations, labeled (P), is that of minimizing the functionalover some class X of functions x and subject to the constraints x(a) = xa, x(b) = xb. Let us take for now the class X of functions to be the continuously differentiable mappings from [a, b] to Rn; we call such functions smooth arcs.

On Asymmetric Distances

2013 ◽  
Vol 1 ◽  
pp. 200-231 ◽  
Author(s):  
Andrea C.G. Mennucci
Keyword(s):  
Total Variation ◽  
Calculus Of Variations ◽  
Distance Function ◽  
Metric Space ◽  
Metric Spaces ◽  
Geodesic Segment ◽  
Intrinsic Metric ◽  
Typical Application ◽  
Variation Formula

Abstract In this paper we discuss asymmetric length structures and asymmetric metric spaces. A length structure induces a (semi)distance function; by using the total variation formula, a (semi)distance function induces a length. In the first part we identify a topology in the set of paths that best describes when the above operations are idempotent. As a typical application, we consider the length of paths defined by a Finslerian functional in Calculus of Variations. In the second part we generalize the setting of General metric spaces of Busemann, and discuss the newly found aspects of the theory: we identify three interesting classes of paths, and compare them; we note that a geodesic segment (as defined by Busemann) is not necessarily continuous in our setting; hence we present three different notions of intrinsic metric space.

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.

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