MATH 402 - Assignment #2
Due on Monday February 7, 2011
Name
Student number
1
Problem 1:
We say that the function f (x1 , . . . , xn ) is positive-homogenous of degree
k in x1 , . . . , xn if
f (x1 , . . . , xn ) = k f (x1 , . . . , xn )
for every > 0.
M402(201) SolutionsAssignment 9
1. The line through (0, ) and (1 , 0) has slope /(1 ). If the axes on which the nine lines are
drawn are labelled (t, x) as usual, then that line is the graph of the function
1
def
t x(t; ) = +
t.
To nd the envelope of thes
M402(201) SolutionsAssignment 10
1. (a) Suppose a minimizing arc exists; call it x. Then there must be constants 0 cfw_0, 1 and
R, not both zero, such that x is extremal for
def
L(t, x, v) = 0
1 + v 2 + x.
Abnormal Case: If 0 = 0, then the Euler-Lagrange
M402(201) SolutionsAssignment 8
1. Arc length ds in R3 obeys (ds)2 = (dx)2 + (dy)2 + (dz)2 . Parametrize the unknown curve from
(0, 0, 0) to (1, 1, 1) as x = t, y = x2 = t2 , and z = z(t): then the problem is to nd the function
z P WS[0, 1] satisfying z(0
UBC Mathematics 402(201)Assignment 10
Due at 09:00 on Thursday 26 March 2015
1. Consider this isoperimetric problem on a xed interval, assuming > 0:
2
x
2
1 + x(t)2 dt : x(0) = 0 = x(2),
min
0
x(t) dt = .
0
(a) Prove: If a minimizing arc exists, its graph
UBC Mathematics 402(201)Assignment 11
Due at 09:00 on Thursday 02 April 2015
1. Consider the free-right-endpoint problem
b
min
b>a, x
f (t, x(t)
1 + x2 (t) dt : x(a) = A, x(b) = g(b) ,
a
where f > 0 and g are given C 1 functions. Use the transversality co
M402(101) SolutionsAssignment 7
1. Here L = v 2 + 2xv 16x2 , so Lv = 2v + 2x and Lx = 2v 32x. It follows that Lvv = 2 > 0
everywhere in (t, x, v)-space, so every extremal in this problem is a smooth solution of (DEL), i.e.,
x(t) + 16x(t) = 0,
0 < t < b.
F
UBC Mathematics 402(201)Assignment 9
Due at 09:00 on Thursday 19 March 2015
1. Carefully sketch the nine lines in the rst quadrant joining (0, ) to (1 , 0), where =
0.1, 0.2, . . . , 0.9. The picture suggests that the family of all such lines, with taking
UBC Mathematics 402(201)Assignment 8
Due at 09:00 on Thursday 12 March 2015
1. A taut string joins the points (0, 0, 0) and (1, 1, 1) on the three-dimensional surface y = x2 . What
curve does the string describe?
2. A heavy gun is installed at the point o
M402(201) SolutionsAssignment 11
1. Here L(t, x, v) = f (t, x) 1 + v 2 , so
Lv = f (t, x)
v
1 + v2
and
1 + v2
L Lv v = f (t, x)
f (t, x)
v
=
.
2
1+v
1 + v2
The endpoint condition x(b) = g(b), with b > a free, requires the point (b, x(b) to lie on the c
M402(101) SolutionsAssignment 3
1. Let t measure distance along the ground from the take-o point (say t = 0) to the
landing point (say t = D). If the aircraft follows the path z = x(t) above the t-axis,
then the innitesimal cost accumulated over a time in
M402(201) SolutionsAssignment 11
1. Here L(t, x, v) = f (t, x) 1 + v 2 , so
Lv = f (t, x)
v
1 + v2
and
1 + v2
L Lv v = f (t, x)
f (t, x)
v
=
.
2
1+v
1 + v2
The endpoint condition x(b) = g(b), with b > a free, requires the point (b, x(b) to lie on the c
M402(201) SolutionsAssignment 10
1. (a) Suppose a minimizing arc exists; call it x. Then there must be constants 0 cfw_0, 1 and
R, not both zero, such that x is extremal for
def
L(t, x, v) = 0
1 + v 2 + x.
Abnormal Case: If 0 = 0, then the Euler-Lagrange
M402(201) SolutionsAssignment 9
1. The line through (0, ) and (1 , 0) has slope /(1 ). If the axes on which the nine lines are
drawn are labelled (t, x) as usual, then that line is the graph of the function
1
def
t x(t; ) = +
t.
To nd the envelope of thes
M402(201) SolutionsAssignment 8
1. Arc length ds in R3 obeys (ds)2 = (dx)2 + (dy)2 + (dz)2 . Parametrize the unknown curve from
(0, 0, 0) to (1, 1, 1) as x = t, y = x2 = t2 , and z = z(t): then the problem is to nd the function
z P WS[0, 1] satisfying z(0
M402(101) SolutionsAssignment 7
1. Here L = v 2 + 2xv 16x2 , so Lv = 2v + 2x and Lx = 2v 32x. It follows that Lvv = 2 > 0
everywhere in (t, x, v)-space, so every extremal in this problem is a smooth solution of (DEL), i.e.,
x(t) + 16x(t) = 0,
0 < t < b.
F
M402(101) SolutionsAssignment 6
1. Rearrange the dierential equation to read u(t) = x(t) + x(t) in order to recognize
this problem as a version of the basic problem in the calculus of variations, namely,
1
min
2
N (t) [x(t) + x(t)] dt : x(0) = A, x(1) = B
M402(101) SolutionsAssignment 4
1. (a) Consider the functional : P WS[a, b] R dened by = 0 + 1 + 2 , where
t1
0 [x] =
b
L(t, x(t), x(t) dt,
1 [x] =
L(t, x(t), x(t) dt,
a
2 [x] = f (x(t1 ).
t1
For any arcs x and h P WS[a, b], we can use results proved in c
M402(101) SolutionsAssignment 5
1. The given subgradient inclusions are, by denition, equivalent to the general inequalities
x x(t)
L(x(t), x(t) + [p(t) p(t)]
L(x, v),
v x(t)
(x(a), x(b) + [p(a)
p(b)]
(x, v),
x x(a)
(x, y),
y x(b)
(1)
(x, y).
(2)
Given
M402(101) SolutionsAssignment 2
1. (a) Here L = 2x2 + t2 v 2 , so
Lv = 2t2 v,
Lx = 4x,
Lvv = 2t2 .
Since L C 2 and Lvv > 0 for all v R and all (t, x) of interest, every extremal must be a C 2
solution of (DEL):
d
2t2 x(t) = 4x(t),
dt
t2 x(t) + 2tx(t) 2x(t
M402(101) SolutionsAssignment 6
1. Rearrange the dierential equation to read u(t) = x(t) + x(t) in order to recognize
this problem as a version of the basic problem in the calculus of variations, namely,
1
min
2
N (t) [x(t) + x(t)] dt : x(0) = A, x(1) = B
UBC Mathematics 402(101)Assignment 6
Due in class on Wednesday 25 February 2015
1. Given a function N continuous and positive-valued on [0, 1], nd the integrable function u: [0, 1] R that gives the minimum value to the integral
1
N (t)u(t)2 dt,
0
subject
This examination has 5 questions on 5 pages.
The University of British Columbia
Final ExaminationsDecember 2007
Mathematics 402
Calculus of Variations (Professor Loewen)
Duration:
Permitted:
Forbidden:
Presentation:
150 minutes.
Any handwritten or instruc
MATH 402 - Assignment #1
Due on Friday January 21, 2011
Name
Student number
1
Problem 1:
(a) Construct a non-negative smooth function which has its support
inside the interval [0, 1].
(b) Construct a non-negative smooth function which has its support
in
UBC Mathematics 402(101)Assignment 9
Due in class on Monday 3 December 2007
1. Write and hand in a computer program that uses dynamic programming to solve one-dimensional
problems in the calculus of variations. The program should use equally spaced partit
MATH 402 - Assignment #3
Due on Monday February 21, 2011
Name
Student number
1
Problem 1:
Consider a pendulum consisting of a bob of mass m and negligible physical extension on the end of a massless rigid rod of length l, suspended
below a xed pivot.
(a
MATH 402 - Assignment #4
Due on Monday March 7, 2011
Name
Student number
1
Problem 1:
Prove that the set of coecients An , Bn in the general solution for the
vibrating string are evaluated as
L
1
n wt (x, 0) dx,
n 0
0
where the initial shape w(x, 0) and