UBC Mathematics 402(101)—Assignment 7
Due in class on Wednesday 04 March 2015
1. Consider the problem below, in which the integration interval [0, b] is fixed:
(
)
Z b
2
2
min Λ[x] :=
ẋ + 2xẋ − 16x dt : x(0) = 0, x(b) = 0 .
0
For which values of b > 0, and which extremals, does Jacobi’s necessary condition hold?
2. Consider the problem below, in which the integration interval [1, b] is fixed:
)
(
Z b
2
1
5x
dt : x(1) = 0, x(b) = 0 .
min Λ[x] :=
ẋ2 − 2
t
t
1
For which values of b > 1, and which admissible extremals, does Jacobi’s necessary condition hold?
3. Consider the basic problem
(Z
)
b
min
t2 ẋ(t)2 + 2βtx(t)ẋ(t) + γx(t)2 dt : x(a) = A, x(b) = B ,
x∈P WS[a,b]
a
p
in which 0 < a < b are fixed. Prove: if γ − β < −1/4 and b > a exp π/ β − γ − 1/4 , then this
problem has no minimum for any A and B.
4. (Ruling out conjugate points by inspection.) Prove that in either of the situations below, the
interval [a, b] contains no points conjugate to a relative to the smooth extremal arc x
b:
(i) The integrand L = L(t, v) is independent of x and of class C 3 , and one has Lvv (t, x(t))
ḃ
>0
for all t in [a, b].
(ii) The integrand L = L(x, v) is independent of t and of class C 3 , and one has
b vv (t) > 0,
L
b xx (t) ≥ 0,
L
b xv (t) = 0,
and L
∀t ∈ [a, b].
[Hint: Let y satisfy Jacobi’s equation with initial conditions y(a) = 0, ẏ(a) = 1. Explain why
b vv (t)ẏ(t) must stay positive on a large interval of the form [a, r).]
L
b Define
5. Let g(· ; α, β) be an extremal for each (α, β), and consider x
b(t) := g(t; α
b, β).
#
"
b gβ (a; α
b
gα (a; α
b, β)
b, β)
.
D(t) = det
b
b
gα (t; α
b, β)
gβ (t; α
b, β)
Suppose that D(a) = 0, but D(t) 6= 0 for each t in some interval of the form (a, a + δ), with δ > 0.
and suppose that in a neighbourhood of t = a, D(t) vanishes only at a. (The two-parameter family
g(t; α, β) is then said to provide a complete solution of the Euler-Lagrange equation near t = a.)
Prove: if D(c) = 0 for some c > a, then c is conjugate to a relative to x
b. (You may assume that g
b vv (t) > 0 for each t.)
is C 3 in (t, α, β), that L is C 3 , and that L
File “hw07”, version of 27 Feb 2015, page 1.
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