Main Article Content
Abstract
Suppose we consider P and Q have coordinates 1 1 (x , y ) , 2 2 (x , y ) and we also
consider the classes of y = y(x) functions , that the boundary conditions 1 1 y(x ) = y
and 2 2 y(x ) = y they are true and Its curves should joins P andQ points. In that case
the purpose of finding a function of this class to minimize the integral
2
1
( ) ( , , ')
x
x
I y = f x y y dx . It should be noted this review only shows that if I (y) a
steady value has. In that case the corresponding steady curve should be a straight line.
But, we know from geometry that I (y) doesn’t have maximizing curve but a minimizing
curve has. So we conclude that it is practically the shortest connecting curve of two points.
Keywords
Article Details
Copyright (c) 2024 Reserved for Kabul University.
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
References
- Abraham, R. and Marsden, j. foundations of mechanics,1985. 2nd ed., Benjamin/Cummings puble.com.
- Anco, s.c.and Bluman, G.W., “Derivation of conservation laws from nonlocal symmetries of Differential Equation,” J. Math. Phys ,1996. 37no. 5 pp. 2361-2375.
- Bernstein, S.N., “sur les equations du Calculus des variation “Ann. Sci Ecolenorm, sup ,1912. 29, pp.431-585.
- Birkhoff, G. and Rota, G., ordinary Differential Equation, 1989. 4th ed., john Wily and sons.
- Bloze, G.A., Lecture on the Calculus of variation, 1931. G.E stechert. and co.
- Browder, F., ed. Mathematical Development Arising from Hilbert Problems, Proceedings of the symposium in pure mathematics 1976. of the American Mathematical Society, vol. 28.
- Caratheodory, C., calculus of variations and partial Differential Equations, 2008. of the first order, Chelsea.
- Courant, R. and Hilbert., D., Methods of mathematical Physics, 1953. vol. 1, john Wiley and sons.
- Forsyth, A.R., calculus of variation with applications, 1987. Dover.
- Giaquinta, M. and Hildebrandt, S., Calculus of variations II, 1996. the lagrangian formalism, springer-verlag.
- Kalnins, E, G., separation of Variables for Riemannian Spaces of constant curvature, 2013. Pitman Monograph, longman.
- Kreyszig, E., Advanced Engineering Mathematics, 1979. 4th ed., University of Toronto press.
References
Abraham, R. and Marsden, j. foundations of mechanics,1985. 2nd ed., Benjamin/Cummings puble.com.
Anco, s.c.and Bluman, G.W., “Derivation of conservation laws from nonlocal symmetries of Differential Equation,” J. Math. Phys ,1996. 37no. 5 pp. 2361-2375.
Bernstein, S.N., “sur les equations du Calculus des variation “Ann. Sci Ecolenorm, sup ,1912. 29, pp.431-585.
Birkhoff, G. and Rota, G., ordinary Differential Equation, 1989. 4th ed., john Wily and sons.
Bloze, G.A., Lecture on the Calculus of variation, 1931. G.E stechert. and co.
Browder, F., ed. Mathematical Development Arising from Hilbert Problems, Proceedings of the symposium in pure mathematics 1976. of the American Mathematical Society, vol. 28.
Caratheodory, C., calculus of variations and partial Differential Equations, 2008. of the first order, Chelsea.
Courant, R. and Hilbert., D., Methods of mathematical Physics, 1953. vol. 1, john Wiley and sons.
Forsyth, A.R., calculus of variation with applications, 1987. Dover.
Giaquinta, M. and Hildebrandt, S., Calculus of variations II, 1996. the lagrangian formalism, springer-verlag.
Kalnins, E, G., separation of Variables for Riemannian Spaces of constant curvature, 2013. Pitman Monograph, longman.
Kreyszig, E., Advanced Engineering Mathematics, 1979. 4th ed., University of Toronto press.