Examples of nonlinear equations
one-dimensional equations
x2 6x + 9 = 0
x cos( x ) = 0
Roots of
Nonlinear Equations
exp( x ) ln( x 2 ) x cos( x ) = 0
two-dimensional equation
version: January 26, 2010
y 2 (1 x ) = x 3
2
2
x + y =1
1
2
Solving nonlinear
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment Four, Due Friday, October 21, 5:00 pm.
[1.] In understanding the behavior of spin1/2 particles in quantum mechanics, you Will
encounter the 2 x 2 Pauli matrices
_ 0 1 _ 0 2' _ 1 0
at 1 0 y z' 0 72 0
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment One, Due Friday, September 30, 5:00 pm.
[1.] We solved the damped harmonic oscillator in class. Do the same to nd the charge
Q(t) on the capacitor plate in an LRC circuit. How do you incorporate the
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment Five, Due Friday, October 28, 5:00 pm.
[1.] In class we solved the problem of two equal masses m1 2 m2 connected by a spring of
force constant k. Generalize the solution to m1 75 m2. Write equations
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment Two, Due Friday, October 7, 5:00 pm.
[1.] Compute cos(i7r).
[2.] A special case of problem 4 of assignment 1 is that multiplication by 2' rotates a complex
number by 90 without changing its length. U
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment Three, Due Friday, October 14, 5:00 pm.
[1.] The Fermi function f (E) = 1/ (65E + 1) plays a central role in the description of
electrons in solids. It gives the number of electrons in a state of ene
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PHYSICS 104A, FALL 2015
MATHEMATICAL PHYSICS
Midterm Exam
I Ital: "I nd = in! mulls are u
' . may 30% Mn your abilities
. a; a tench-r
[1.] Write down the solutions to the equation 26 = 2.
[2.] In. understanding the behavior of spin1 particles in quan
Examples of nonlinear equations
Simple harmonic oscillator (linear ODE)
Nonlinear Differential Equations
d 2 x (t )
= kx(t )
dt
m
More complicated motion (nonlinear ODE)
and The Beauty of Chaos
m
d 2 x (t )
= kx (t )(1 x (t )
dt
Other examples: weather pa
Modeling of Data
Modeling of Data
5
5
4
4
Interpolation
Data fit
3
5
Y
Y
3
2
2
data points
linear interpolation
spline interpolation
4
1
data poits
polinomial fit
4
1
3
3
0
2
4
6
8
10
X
Interpolation =
local approximation
0
Y
2
4
6
8
10
2
X
2
Data modelin
subroutine pde01(f,dx,dy,imax,jmax,iter,tol,omega) ! Laplace equation solver with Dirichlet BCs ! f(i,j) function f(i,j) ! dx, dy grid increments ! imax number of grid points in the x direction ! jmax number of grid points in the y direction ! iter maximu
Random Processes
Random or Stochastic processes
You cannot predict from the observation of one event,
how the next will come out
Examples:
Coin: the only prediction about outcome
50% the coin will land on its tail
Dice: In large number of throws
probabi
The motion of the Planets
Gravitational force
Gravitational force is one of the four fundamental forces
r
mM r
mM
F = G 2 r = G 3 r
r
r
m3
11
G = 6.67 10
kg s 2
1/20
2/20
See www.nasa.gov for more information
Simple case: motion in a central field (m<M)
E
Newtons Laws
Newtons first law
In the absence of forces, a particle moves with constant
velocity.
Motion of Projectiles
and
Charged Particles
Newtons second law
For any particle of mass m, the net force F on the particle is
always equal to the mass m time
Machine representation and
precision
Machine
Every computer has a limit how small or large a
number can be
precision
A computer represent numbers in the binary form.
Word length: number of bytes used to store a number
Most common architecture:
Word length
Differential Equations
Examples:
Most fundamental and basic equations in physics
as well as frequently occurring problems appear as
Simple harmonic oscillator
differential equations.
m
d 2 x (t )
= kx (t )
dt 2
Schrdinger equation (example for 1D)
ih
d (
Computational Projects
1
2
Art and Science
Milestones
Computational Physics is an art
(requires imagination and creativity)
and science
(uses specific methods and techniques)
1.
Problem definition
Problem analysis
3. Equations and data
4. Computational pr
Short introduction to C+
C+
short reference
for
introductory computational physics
Reference books
Structure of a program
Variables, Data Types, and Constants
Operators
Basic Input/Output
Control Structures
Functions
Arrays
Input/Output with files
Pointer
PHYSICS 104A, FALL 2015
MATHEMATICAL PHYSICS
Assignment Six, Due Friday, November 11, 5:00 pm.
[1.] In the usual basis
1
e1 = 0
0
0
e2 = 1
0
0
e3 = 0
1
a vector ~v has the components given below, and an operator O is represented by the matrix
als
PHYSICS 104A, FALL 2016
MATHEMATICAL PHYSICS
Assignment One, Due Friday, October 7, 5:00 pm.
[1.] Compute cos(i).
[2.] A special case of problem 4 of assignment 1 is that multiplication by i rotates a complex
number by 90 without changing its length. Use
PHYSICS 104A, FALL 2015
MATHEMATICAL PHYSICS
Assignment Seven, Due Monday, November 21, 5:00 pm.
[1.] Do the Fourier decompositions of the periodic functions shown in the figures.
[2.] Rectification of V (t) preserves unchanged any positive voltages, but