(a) The Euler-Lagrange equations for the given functional are: For y₁: - = 2 2 d/dx(ǝL/ǝy₁') - ƏL/Əy₁ = 0 ⇒ d/dx(14y₁' + 24y2') - (14y₁ + 24y2) = 0 ⇒ 14y₁" + 24y2" - 14y₁ - 24y2 =0 ⇒ 14y₁" - 14y₁ + 24y2" - 24y2 = 0 For y₂: d/dx(ǝL/ǝy2') - ƏL/ǝy2 = 0 => 22 d/dx(24y₁' + 42y2') - (24y₁ + 42y2 + 12) = 0 => 24y1" + 42y2" - 24y₁ - 42y2 - 12 = 0 how come?? 2 2 (b) Let's substitute y₁ = az₁z₂ and y₂ = z₁² + z2² into the functional S: S[Z1, Z2] = √ dx [7(az₁z2)² + 24(az₁z₂)(z₁ ² + Z₂²) + 21(z₁² + z₂²)² + 4(Z₁² + Z₂²) + 12(ɑz₁Z2)(Z₁² + Z₂²) + 9(z₁² + z₂²)] 2 1 -2 4 2 = √ dx [7ɑ²z₁²z₂² + 24ɑ(z₁³z₂ + Z₁Z₂³) + 21(z₁ª + 2z₁²z₂² + Z₂²) + 4(Z₁² + Z₂²) + 12α(z₁³z₂ + Z₁Z₂³) + 9(z₁² + z₂²)] 2 To separate this into S₁[z₁] and S2[22], we need to choose a such that the mixed terms (z₁³z₂ and z₁Z2³) vanish. 24a+12a=0 → 36α = 0 0 = p = = With a 0, we get: 4 2. 2 2 2 S[Z1, Z2] = √ dx [21(z₁₁ + 2z₁ ²z₂² + z₂^) + 4(z₁² + z₂²) + 9(z₁² + z₂²)] 4 4 2 2 = √ dx [21z₁² + 42z₁²z₂² + 21z₂² + 4z₁² + 4z₂² + 9z₁²+9z22] 4 = √ dx [(21z₁² + 13z₁²) + (21z2ª + 55z2²)] 4 4 So, S₁[z₁] = √ dx (21z₁₁ + 13z₁²) and S₂[72] = √ dx (21z2ª + 55z2²). Consider the functional (a) S[y₁, ¥2] = [ dx [7yf² + 24y{y2 + 21y2² + 4y{ + 12y1v2 + 9v3] . two coupled Euler-Lagrange equations giving the stationary paths of S[y1, y2]. (b) Express the functional S in terms of the two new dependent variables 21, 22, where y₁ and y2 are given in terms of the new dependent variables by Y₁ = azı - 322, Y1 Y2 = 21+ 222, and where a is a constant. Determine the value of a such that the functional S[21, 22] can be written as the sum of two functionals, S₁[21] + S2[22], where S₁[21] depends only upon 21 and S2[22] depends only on upon z₂. Calculate S₁[21] and S2[22] for this value of a.
(a) The Euler-Lagrange equations for the given functional are: For y₁: - = 2 2 d/dx(ǝL/ǝy₁') - ƏL/Əy₁ = 0 ⇒ d/dx(14y₁' + 24y2') - (14y₁ + 24y2) = 0 ⇒ 14y₁" + 24y2" - 14y₁ - 24y2 =0 ⇒ 14y₁" - 14y₁ + 24y2" - 24y2 = 0 For y₂: d/dx(ǝL/ǝy2') - ƏL/ǝy2 = 0 => 22 d/dx(24y₁' + 42y2') - (24y₁ + 42y2 + 12) = 0 => 24y1" + 42y2" - 24y₁ - 42y2 - 12 = 0 how come?? 2 2 (b) Let's substitute y₁ = az₁z₂ and y₂ = z₁² + z2² into the functional S: S[Z1, Z2] = √ dx [7(az₁z2)² + 24(az₁z₂)(z₁ ² + Z₂²) + 21(z₁² + z₂²)² + 4(Z₁² + Z₂²) + 12(ɑz₁Z2)(Z₁² + Z₂²) + 9(z₁² + z₂²)] 2 1 -2 4 2 = √ dx [7ɑ²z₁²z₂² + 24ɑ(z₁³z₂ + Z₁Z₂³) + 21(z₁ª + 2z₁²z₂² + Z₂²) + 4(Z₁² + Z₂²) + 12α(z₁³z₂ + Z₁Z₂³) + 9(z₁² + z₂²)] 2 To separate this into S₁[z₁] and S2[22], we need to choose a such that the mixed terms (z₁³z₂ and z₁Z2³) vanish. 24a+12a=0 → 36α = 0 0 = p = = With a 0, we get: 4 2. 2 2 2 S[Z1, Z2] = √ dx [21(z₁₁ + 2z₁ ²z₂² + z₂^) + 4(z₁² + z₂²) + 9(z₁² + z₂²)] 4 4 2 2 = √ dx [21z₁² + 42z₁²z₂² + 21z₂² + 4z₁² + 4z₂² + 9z₁²+9z22] 4 = √ dx [(21z₁² + 13z₁²) + (21z2ª + 55z2²)] 4 4 So, S₁[z₁] = √ dx (21z₁₁ + 13z₁²) and S₂[72] = √ dx (21z2ª + 55z2²). Consider the functional (a) S[y₁, ¥2] = [ dx [7yf² + 24y{y2 + 21y2² + 4y{ + 12y1v2 + 9v3] . two coupled Euler-Lagrange equations giving the stationary paths of S[y1, y2]. (b) Express the functional S in terms of the two new dependent variables 21, 22, where y₁ and y2 are given in terms of the new dependent variables by Y₁ = azı - 322, Y1 Y2 = 21+ 222, and where a is a constant. Determine the value of a such that the functional S[21, 22] can be written as the sum of two functionals, S₁[21] + S2[22], where S₁[21] depends only upon 21 and S2[22] depends only on upon z₂. Calculate S₁[21] and S2[22] for this value of a.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![(a) The Euler-Lagrange equations for the given functional are:
For y₁:
-
= 2 2
d/dx(ǝL/ǝy₁') - ƏL/Əy₁ = 0
⇒ d/dx(14y₁' + 24y2') - (14y₁ + 24y2) = 0
⇒ 14y₁" + 24y2" - 14y₁ - 24y2 =0
⇒ 14y₁" - 14y₁ + 24y2" - 24y2 = 0
For y₂:
d/dx(ǝL/ǝy2') - ƏL/ǝy2 = 0
=>
22
d/dx(24y₁' + 42y2') - (24y₁ + 42y2 + 12) = 0
=>
24y1" + 42y2" - 24y₁ - 42y2 - 12 = 0
how come??
2
2
(b) Let's substitute y₁ = az₁z₂ and y₂ = z₁² + z2² into the functional S:
S[Z1, Z2] = √ dx [7(az₁z2)² + 24(az₁z₂)(z₁ ² + Z₂²) + 21(z₁² + z₂²)² + 4(Z₁² + Z₂²) +
12(ɑz₁Z2)(Z₁² + Z₂²) + 9(z₁² + z₂²)]
2
1 -2
4
2
= √ dx [7ɑ²z₁²z₂² + 24ɑ(z₁³z₂ + Z₁Z₂³) + 21(z₁ª + 2z₁²z₂² + Z₂²) + 4(Z₁² + Z₂²) +
12α(z₁³z₂ + Z₁Z₂³) + 9(z₁² + z₂²)]
2
To separate this into S₁[z₁] and S2[22], we need to choose a such that the
mixed terms (z₁³z₂ and z₁Z2³) vanish.
24a+12a=0
→ 36α = 0
0 = p =
=
With a 0, we get:
4
2. 2
2
2
S[Z1, Z2] = √ dx [21(z₁₁ + 2z₁ ²z₂² + z₂^) + 4(z₁² + z₂²) + 9(z₁² + z₂²)]
4
4
2
2
= √ dx [21z₁² + 42z₁²z₂² + 21z₂² + 4z₁² + 4z₂² + 9z₁²+9z22]
4
= √ dx [(21z₁² + 13z₁²) + (21z2ª + 55z2²)]
4
4
So, S₁[z₁] = √ dx (21z₁₁ + 13z₁²) and S₂[72] = √ dx (21z2ª + 55z2²).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F20989cb3-1e4a-418f-ad9b-84f25b37b83b%2Ftjly3_processed.png&w=3840&q=75)
Transcribed Image Text:(a) The Euler-Lagrange equations for the given functional are:
For y₁:
-
= 2 2
d/dx(ǝL/ǝy₁') - ƏL/Əy₁ = 0
⇒ d/dx(14y₁' + 24y2') - (14y₁ + 24y2) = 0
⇒ 14y₁" + 24y2" - 14y₁ - 24y2 =0
⇒ 14y₁" - 14y₁ + 24y2" - 24y2 = 0
For y₂:
d/dx(ǝL/ǝy2') - ƏL/ǝy2 = 0
=>
22
d/dx(24y₁' + 42y2') - (24y₁ + 42y2 + 12) = 0
=>
24y1" + 42y2" - 24y₁ - 42y2 - 12 = 0
how come??
2
2
(b) Let's substitute y₁ = az₁z₂ and y₂ = z₁² + z2² into the functional S:
S[Z1, Z2] = √ dx [7(az₁z2)² + 24(az₁z₂)(z₁ ² + Z₂²) + 21(z₁² + z₂²)² + 4(Z₁² + Z₂²) +
12(ɑz₁Z2)(Z₁² + Z₂²) + 9(z₁² + z₂²)]
2
1 -2
4
2
= √ dx [7ɑ²z₁²z₂² + 24ɑ(z₁³z₂ + Z₁Z₂³) + 21(z₁ª + 2z₁²z₂² + Z₂²) + 4(Z₁² + Z₂²) +
12α(z₁³z₂ + Z₁Z₂³) + 9(z₁² + z₂²)]
2
To separate this into S₁[z₁] and S2[22], we need to choose a such that the
mixed terms (z₁³z₂ and z₁Z2³) vanish.
24a+12a=0
→ 36α = 0
0 = p =
=
With a 0, we get:
4
2. 2
2
2
S[Z1, Z2] = √ dx [21(z₁₁ + 2z₁ ²z₂² + z₂^) + 4(z₁² + z₂²) + 9(z₁² + z₂²)]
4
4
2
2
= √ dx [21z₁² + 42z₁²z₂² + 21z₂² + 4z₁² + 4z₂² + 9z₁²+9z22]
4
= √ dx [(21z₁² + 13z₁²) + (21z2ª + 55z2²)]
4
4
So, S₁[z₁] = √ dx (21z₁₁ + 13z₁²) and S₂[72] = √ dx (21z2ª + 55z2²).
![Consider the functional
(a)
S[y₁, ¥2] = [ dx [7yf² + 24y{y2 + 21y2² + 4y{ + 12y1v2 + 9v3] .
two coupled Euler-Lagrange equations giving the
stationary paths of S[y1, y2].
(b) Express the functional S in terms of the two new dependent
variables 21, 22, where y₁ and y2 are given in terms of the new
dependent variables by
Y₁ = azı - 322,
Y1
Y2 = 21+ 222,
and where a is a constant.
Determine the value of a such that the functional S[21, 22] can be
written as the sum of two functionals, S₁[21] + S2[22], where S₁[21]
depends only upon 21 and S2[22] depends only on upon z₂.
Calculate S₁[21] and S2[22] for this value of a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F20989cb3-1e4a-418f-ad9b-84f25b37b83b%2Fsywden5_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the functional
(a)
S[y₁, ¥2] = [ dx [7yf² + 24y{y2 + 21y2² + 4y{ + 12y1v2 + 9v3] .
two coupled Euler-Lagrange equations giving the
stationary paths of S[y1, y2].
(b) Express the functional S in terms of the two new dependent
variables 21, 22, where y₁ and y2 are given in terms of the new
dependent variables by
Y₁ = azı - 322,
Y1
Y2 = 21+ 222,
and where a is a constant.
Determine the value of a such that the functional S[21, 22] can be
written as the sum of two functionals, S₁[21] + S2[22], where S₁[21]
depends only upon 21 and S2[22] depends only on upon z₂.
Calculate S₁[21] and S2[22] for this value of a.
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