Let n ≥ 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors, and consider the functional of n dependent variables x1 In: rt1 C[x] = [*^ dt L(t,x,x), x(to) = xo, x(t₁) = x1, where x = to ..., 1, . . ., £n) and x = (±1,...,xn), and where ik (x1,..., k = 1, . . ., n. Show that on a stationary path = dxk for dt n d ᎥᏞ aL L ik == dt Əxk Ət k=1 Hence find a first-integral for the case when L is independent of t.
Let n ≥ 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors, and consider the functional of n dependent variables x1 In: rt1 C[x] = [*^ dt L(t,x,x), x(to) = xo, x(t₁) = x1, where x = to ..., 1, . . ., £n) and x = (±1,...,xn), and where ik (x1,..., k = 1, . . ., n. Show that on a stationary path = dxk for dt n d ᎥᏞ aL L ik == dt Əxk Ət k=1 Hence find a first-integral for the case when L is independent of t.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Let n ≥ 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors,
and consider the functional of n dependent variables x1 In:
rt1
C[x] = [*^ dt L(t,x,x), x(to) = xo, x(t₁) = x1,
where x =
to
...,
1, . . ., £n) and x = (±1,...,xn), and where ik
(x1,...,
k = 1, . . ., n.
Show that on a stationary path
=
dxk for
dt
n
d
ᎥᏞ
aL
L
ik
==
dt
Əxk
Ət
k=1
Hence find a first-integral for the case when L is independent of t.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2Fd6ffecc7-6772-46dd-a49a-16c0d2379985%2Fse7ttdo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let n ≥ 1 be an integer, let to < t₁ and x0, x₁ be real constant n-vectors,
and consider the functional of n dependent variables x1 In:
rt1
C[x] = [*^ dt L(t,x,x), x(to) = xo, x(t₁) = x1,
where x =
to
...,
1, . . ., £n) and x = (±1,...,xn), and where ik
(x1,...,
k = 1, . . ., n.
Show that on a stationary path
=
dxk for
dt
n
d
ᎥᏞ
aL
L
ik
==
dt
Əxk
Ət
k=1
Hence find a first-integral for the case when L is independent of t.
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