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
MATH 402 - Assignment #5
Due on Monday March 28, 2011
Name
Student number
1
Problem 1:
(a) Find all possible piecewise smooth extremals of the functional
4
(y
2
1)2 dx,
y(0) = 0,
y(4) = 2,
0
which
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
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 fu
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
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 pr
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
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 a
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 s
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
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
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-axi
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
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 fu
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 pr
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 s
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, name
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
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
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
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 stri
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, name
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 mi
UBC Mathematics 402(101)Assignment 6
Due in class on Monday 5 November 2007
1. Consider the following problem:
2
min
0
4
x(t)2 + x(t)2 (x(t) t)2 x(t)3 + 2tx(t)2 dt : x(0) = 0, x(2) = e .
3
(a) Find al
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
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
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 v
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