Newtons Laws
Projectiles and Charged Particles
Newtons first law In the absence of forces, a particle moves with constant velocity. Newtons second law For any particle of mass m, the net force F on the particle is always equal to the mass m times the part
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
See www.nasa.gov for more information
Simple case: motion in a central field (m<M)
Equation of
What is Dislin?
Dislin is a high-level plotting library for displaying data as curves, bar graphs, pie charts, 3D-colour plots, surfaces, contours and maps. Dislin has been developed by Helmut Michels from Max Planck Institute for Solar System Research, L
Machine
precision
Machine representation and precision
Every computer has a limit how small or large a number can be A computer represent numbers in the binary form. Word length: number of bytes used to store a number Most common architecture: Word length
Short introduction to C+
C+
short reference for introductory computational physics
Structure of a program Variables, Data Types, and Constants Operators Basic Input/Output Control Structures Functions Arrays Input/Output with files Pointers Classes
Refe
in scientific writing and publishing
Alexander Godunov Department of Physics, ODU November 2006
Why do I need LaTeX? What is LaTeX? Where can I get LaTeX? How can I use LaTeX? Who can help me with LaTeX?
scientific publishing
scientific publishing
History
Modeling of Data
5 4
3
Interpolation
4
Y
2
1
data points linear interpolation spline interpolation
3
0 0 2 4 6 8 10
X
Y
2 1 0 2 4 6 8 10
Interpolation = local approximation
X
1
Modeling of Data
5 4
3
Data fit
5
Y
2
4 1 3 0 0 2 4 6 8 10
data poits polinomi
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 probabili
Examples of nonlinear equations
Simple harmonic oscillator (linear ODE)
Nonlinear Differential Equations
and The Beauty of Chaos
m
d 2 x (t ) = kx(t ) dt d 2 x (t ) = kx (t )(1 x (t ) dt
More complicated motion (nonlinear ODE)
m
Other examples: weather pa
Differential Equations
Most fundamental and basic equations in physics as well as frequently occurring problems appear as differential equations.
Examples:
Simple harmonic oscillator
m
d 2 x (t ) = kx (t ) dt 2
Schrdinger equation (example for 1D)
ih
d (
Matrices
An mn matrix is a rectangular array of complex or real numbers arranged in m rows and n columns:
Types, Operations, etc.
Types: square, symmetric, diagonal, Hermithean, Basic operations: A+B, A-B, AB (ABBA). Square matrices
Determinant: det(A) In
Integration
S = f ( x )dx
a b
Part 1
Exact integration
Exact integration Simple numerical methods Advanced numerical methods
1
2
Three possible ways for exact integration
Standard techniques of integration substitution rule, integration by parts, using id
Data types in science
Discrete data (data tables)
Interpolation
Experiment Observations Calculations
Continuous data
Analytics functions Analytic solutions
1
2
From continuous to discrete
From discrete to continuous?
?
soon after hurricane Isabel (Septem
Examples of nonlinear equations
one-dimensional equations
Roots of Nonlinear Equations
x2 6x + 9 = 0 x cos( x ) = 0 exp( x ) ln( x 2 ) x cos( x ) = 0
two-dimensional equation
y 2 (1 x ) = x 3 2 2 x + y =1
1 2
1.1 Introduction
Given - a continuous nonline
Linux
Common Linux features
Multi-user (user accounts, multiple users logged in) Multitasking (Servers, daemons) Graphical user interface (X Window system) Hardware support (drivers) Network connectivity Network servers Application support
1
Linux
Short H
Computational Projects
1
2
Art and Science
Computational Physics is an art (requires imagination and creativity) and science (uses specific methods and techniques)
Milestones
Problem definition Problem analysis 3. Equations and data 4. Computational proje
Basics of Computational Physics
What is Computational Physics? Basic computer hardware Software 1: operating systems Software 2: Programming languages Software 3: Problem-solving environment
What does Computational Physics do?
Atomic Physics studies atoms