Solutions to homework assignment #3
Math 119B
UC Davis, Spring 2012
1. Find the stationary points of the following functionals:
(a)
1
0 (q
(x)2 + 12xq (x) dx, q (0) = 0, q (1) = 2
1
Solution. The functional is 0 L(q , q, x) where L(q , q, x) = q 2 + 12xq
Solutions to Midterm Quiz #1
Math 119B
UC Davis, Spring 2012
Question 1
Dene the functional
/2
(q (t)2 q (t)2 ) dt
(q ) =
0
Find the function q where (q ) is minimal, subject to the boundary conditions q (0) = 0, q (/2) = 3.
Solution. The functional is d
Solutions to homework assignment #1
Math 119B
UC Davis, Spring 2012
1. (a) Starting from the Hamiltonian H (p, q, t) = p2 + tq 4 , nd the associated Lagrangian, and
write the associated ODEs in both the Lagrangian and Hamiltonian forms.
Solution. The syst
Solutions to review problems from HW #5
Math 119B (UC Davis, Spring 2012)
7. Use matrix exponentials, or any other method, to nd the solution (x(t), y (t) of the ODE
system
x = 2x + y,
y = x + 2y
satisfying the initial conditions x(0) = 0, y (0) = 1.
Solu
Solutions to nal exam (6/12/12)
Math 119B
UC Davis, Spring 2012
Question 1 (25 points)
Use matrix exponentials, or any other method, to nd the solution (x(t), y (t) of the ODE system
x = 2x y,
y = 2x y,
satisfying the initial conditions x(0) = 0, y (0) =
Solutions to midterm quiz #2
Math 119B
UC Davis, Spring 2012
Question 1
(a) Prove that if A, B are commuting square matrices (i.e., matrices which satisfy AB = BA) then
exp(A + B ) = exp(A) exp(B ).
Solution.
exp(A + B ) =
n=0
=
n=0
1
(A + B )n
n!
1
n!
n
Solutions to homework assignment #2
Math 119B
UC Davis, Spring 2012
1. In the computer game Angry Birds Space, let x(t) = (x(t), y (t) denote the position of
an angry bird at time t ying near a planet centered at (0, 0), and denote r(t) = |x(t)| =
x(t)2 +
Solutions to homework assignment #4
Math 119B
UC Davis, Spring 2012
1. A rope of length and uniform linear density hangs between two points (A/2, 0), (A/2, 0),
where we assume that A. Its shape y = q (x) is determined by the requirement that the
potential
Solutions to homework assignment #5
Math 119B
UC Davis, Spring 2012
1. Find the solution xn to the recurrence
xn+2 = xn+1 + 2xn
satisfying the initial conditions x1 = 5, x2 = 1.
Solution. As in the example of the Fibonacci numbers discussed in class, we c
Solutions to homework Assignment #6
Math 119B
UC Davis, Spring 2012
Problems
1. For each of the following maps acting on the interval [0, 1], sketch their graphs, nd their
xed points and determine for each xed point whether it is asymptotically stable (a.
Solutions to homework assignment #7
Math 119B
UC Davis, Spring 2012
1. A xed point x of an interval map T is called superstable if T (x ) = 0. Find the value of
0 < r 4 for which the logistic map Lr has a superstable xed point.
Solution. We know that the
Solutions to homework assignment #8
Math 119B
UC Davis, Spring 2012
1. A child standing on a swing bends her knees up and down in a periodic motion. This causes a
slight change in the resonant frequency of the swing. In the approximation of small amplitud