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E. our arcs have been described by two variables, namely x, y. A natural generalization is to consider problems involving arcs in three-space with coordinates x, y, z or in even higher dimensional space, say N +1 dimensional space with coordinates x, y1 , · · · , yN . The problem to be considered then involves an integral of the form x2 I= F (x, y1 , · · · , yN , y1 , · · · , yN )dx . (1) x1 and a class of admissible arcs y where superscript bar designates a vector arc, with components y : yi(x) x1 ≤ x ≤ x2 i = 1, · · · , N (2) on which the integral (1) has a well defined value.

E. the integral to be minimized is x2 F (x, y ) dx (16) x1 where F does not contain y explicitly. In this case the first Euler’s equation (12) becomes along an extremal d Fy = 0 (17) dx or Fy = C (18) where C is a constant. This is a first order differential equation which does not contain y. This was the case in the shortest distance problem done before. Case 2 If the integrand does not depend on the independent variable x, i. e. if we have to minimize x2 F (y, y ) dx (19) x1 then the second Euler equation (13) becomes d (F − y Fy ) = 0 dx (20) F − y Fy = C (21) or (where C is a constant) a first order equation.

4 2. Find the extremals for I = 1 1 2 (y ) + yy + y + y dx 2 0 where end values of y are free. 3. Solve the Euler-Lagrange equation for b I = y 1 + (y )2 dx a where y(a) = A, y(b) = B. b. Investigate the special case when a = −b, A=B and show that depending upon the relative size of b, B there may be none, one or two candidate curves that satisfy the requisite endpoints conditions. 4. Solve the Euler-Lagrange equation associated with b I = a y 2 − yy + (y ) 2 dx 5. What is the relevant Euler-Lagrange equation associated with I = 1 0 y 2 + 2xy + (y ) 2 dx 6.

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