[4] An N = 1 dynamical system obeys the equation
du
= ru + 2bu2 u3 ,
dt
where r is a control parameter, and where b > 0 is a constant.
(a) Find and classify all bifurcations for this system.
[7 points]
(b) Sketch the fixed points u versus r.
[6 points]
No
@ﬂogAEME 3
jarvf/O‘Zﬁnr‘a/v Mac/Laynz‘C—S'
, _
“gm—w ‘ 13.5 Hr A head mass m is threaded on a frictionless wire that is bent into a helix with cylindrical polar
coordinates (,0, ¢, 2) satisfying 2 = 0d) and p = R, with c and R constants. The z axis point
Chapter 14
Continuum Mechanics
14.1 Strings
Consider a string of linear mass density (x) under tension (x).1 Let the string move in a plane, such that its shape is described by a smooth function y(x), the vertical displacement of the string at horizontal
Chapter 13
Rigid Body Motion and Rotational Dynamics
13.1 Rigid Bodies
A rigid body consists of a group of particles whose separations are all xed in magnitude. Six independent coordinates are required to completely specify the position and orientation of
Chapter 15
Special Relativity
For an extraordinarily lucid, if characteristically brief, discussion, see chs. 1 and 2 of L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Course of Theoretical Physics, vol. 2).
15.1
Introduction
All distanc
Final
Mar. 17, 2009
PROBLEM (1)
Physics 110B
(4 points)
Assume a neutrino beam is made by generating a beam of + with kinetic energy T = 500 GeV.
The + decay to a positive muon and a neutrino of the muon type, + + . After the + beam
is formed, a long evac
Chapter 16
Dynamical Systems
16.1
16.1.1
Introduction
Phase space and phase curves
Dynamics is the study of motion through phase space. The phase space of a given dynamical system is described as an N -dimensional manifold, M. A (dierentiable) manifold M
Chapter 12
Noninertial Reference Frames
12.1 Accelerated Coordinate Systems
A reference frame which is fixed with respect to a rotating rigid body is not inertial. The parade example of this is an observer fixed on the surface of the earth. Due to the rot
PROBLEM (1)
(10 points)
The n early s pherical p lanet S iloiroc h as a d ay e qual t o 1 0 e arth
h ours, a r adius o f
10,000 k m, a nd a n a cceleration d ue t o g ravity a lone o f.2 m f s 2 a t t he s urface.
a)
F ind t he m agnitude o f t he c entri
Midtern
Feb. 11, 2011
Physics 110B
(6 points)
Secret Number
PROBLEM (1)
Hurricane Katrina was the sixth strongest Atlantic hurricane recorded. As a category 5 hurricane, it had peak winds of 280 km/hr over the gulf of Mexico (25 degrees north latitude ).
Final
Mar. 16, 2011
PROBLEM (1)
Physics 110B
(4 points)
a) Use the operator
=
e ^ xi i
to show that the divergence of a curl is zero. A =0 Ak xi xj
ijk
A =
First interchange the indices i and j in the tensor picking up a minus sign, then change the order
4. The Hamiltonian Formalism
Well now move onto the next level in the formalism of classical mechanics, due initially
to Hamilton around 1830. While we wont use Hamiltons approach to solve any further
complicated problems, we will use it to reveal much mo