Necessary conditions for multiple integral problem in the calculus of variations

1982 ◽  
Vol 260 (2) ◽  
pp. 175-189 ◽  
Author(s):  
Viorel Barbu
Keyword(s):  
Calculus Of Variations ◽  
Necessary Conditions ◽  
Multiple Integral ◽  
Integral Problem ◽  
Multiple Integral Problem

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.

On conservation laws and necessary conditions in the calculus of variations

2002 ◽  
Vol 132 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
G. Francfort ◽  
J. Sivaloganathan
Keyword(s):  
Conservation Laws ◽  
Calculus Of Variations ◽  
Conservation Law ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Integral Functional ◽  
Variational Symmetry

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

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