Materi kuliah civil Tokyo University

  

1	Variational Method
1.1    Stationary problem of a function
Let F (x1; x2; : : : ; xn) be a function of independent variables x1; x2; : : : ; xn.  The problem is to
	Find the values of x1; x2; : : : ; xn for which the function F (x1; x2; : : : ; xn) becomes minimum
	(stationary).
	Consider the small variation of F		at x1; x2; : : : ; xn  as folows:
	±F    =    F (x1 + ±x1; x2 + ±x2; : : : ; xn + ±xn) ¡ F (x1; x2; : : : ; xn)
		@F	@F	@F
	¼	@x1 ±x1 + @x2		±x2 + : : : + @xn ±xn  = 0	(1)

Since x₁; x₂; : : : ; x_n are independent variables, the solution for the above problem can be determined by the following conditions:

		@F	= 0;	@F	= 0; : : : ;  @F   = 0:	(2)
		@x1		@x2	@xn
1.2	Functional

A functional is a function of functions such as F (x; u₁; u₂), where u₁ and u₂ are functions of the variable x. As shown in the sequel, the integral form of a functional defined as

^∫ b

J[u] =       F (x; u; u⁰ )dx                                                       (3)

a

appears in many problems, where u⁰ = du=dx. The function u is determined so that the functional J[u] shows a minimum value.

Example 1  Fermat’s principle:

Find the ray path of the light in an inhomogeneous two dimensional medium with the speed Á(x; u).

This problem is equivalent to determining the function u = u(x) so as to minimize the travelling time T of the light defined by

			p	2	2		x1	p	2
T  = ∫		Á(x;ds u)  = ∫		dx   + du		= ∫x0		Á1(x;+ uu)0     dx		(4)
				Á(x; u)
Example 2  String sustained at two points:
Determine the shape of the uniform string with the length ` sustained at two points
u(x0) = u0  and u(x1) = u1  in gravity field.
The problem is to find the function u = u(x) that minimize the potential energy
x1	p	E = ½g ∫  uds = ½g ∫xx01			u√1 + u02dx					(5)
		2
Example subjected3Shortesttoroute`= ∫onx0	the1 +surface:u0dx.
Determine the shortest curve between two points on the surface defined by x = x(u; v), y = y(u; v),
z = z(u; v) in a three dimensional space.
Let L be the length between two points u = u0  and u = u1  on the curve v = v(u) on the surface
x = x(u; v), y = y(u; v), z = z(u; v) defined by
L = ∫  ds = ∫  √dx2 + dy2 + dz2  = ∫uu0							1  √e + 2fv0  + gv02du			(6)

where we use dx = (@x=@u)du + (@x=@v)(dv=du)du = (x_u + x_vv⁰ )du, dx² = x²_u + x²_vv⁰² and so on. In eq.(6), e; f , and g are defined by

e = x²_u + y_u² + z_u²;  f  = x_ux_v + y_uy_v + z_uz_v;  g = x²_v + y_v² + z_v²:

Then the problem is formulated to find the function v = v(u) so that the length L minimum value.

1

+ 2x_ux_vv⁰

(7) shows the

Example 4  Fastest route for mass sliding

Assume that the mass falls down from the point A (x₀; 0) to the point B (x₁; u₁) along the frictionless slope with zero initial velocity in the two dimensional plane in the gravity

field. Determine the route on the slope where the mass slides fastest. p

Since the velocity v at the position u is given by v = 2gu, the problem is to find the function u that minimize the time T defined by

x1	p		2
T  = ∫x0	1	+ u0		dx	(8)
	p	2gu

As you see, this problem is similar to the example 1.