1. Change of Variable: Let : [to, ti] R be a smooth function on the interval [to, ti] such that '(t) > 0 for all t e [to,ti] and let p(to) = 10, (t1) = x1. Using the transformation r = (t), the functional J defined by %3D %3D %3D J(y) = | f(x, y, y') dæ %3D can be transformed to the functional K defined by K(Y) = F(t, Y,Y) dt, %3D to where, for Y(t) = y((t)), Y denotes dY/dt and %3D

1. Change of Variable: Let : [to, ti] R be a smooth function on the interval [to, ti] such that '(t) > 0 for all t e [to,ti] and let p(to) = 10, (t1) = x1. Using the transformation r = (t), the functional J defined by %3D %3D %3D J(y) = | f(x, y, y') dæ %3D can be transformed to the functional K defined by K(Y) = F(t, Y,Y) dt, %3D to where, for Y(t) = y((t)), Y denotes dY/dt and %3D

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

1. Change of Variable: Let : [to,ti] – R be a smooth function on the
interval [to, ti] such that '(t) > 0 for all t e [to,ti] and let (to) = 20,
(t1) = x1. Using the transformation r =
by
v(t), the functional J defined
J(y) = | f(x, y,y') dr
%3D
can be transformed to the functional K defined by
t1
K(Y) = | F(t, Y,Y) dt,
to
where, for Y(t) = y(4(t)), Y denotes dY/dt and
F(t, Y, Ý) = f(4(t), Y, Ỷ )µ' (t).
Prove by direct calculation that
OF
d af
dr dy'
d OF
%3D
dt aÝ
and hence that
is an extremal for J if and only if Y is an extremal for
K.

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