Lecture 01

Welcome to Physics 331:

Introduction to Numerical Techniques in Physics Instructor: Joaquín Drut

Lecture 1

Logistics…

Phys 331 Introduction to Numerical Techniques in Physics Spring 2016 Course information Instructor: Joaquín E. Drut. Email: drut at email.unc.edu. Office: Phillips 296 Where and When: Class: Phillips 265 - Mo-We-Fr, 12:20pm-1:10pm Mo-We : Lectures Fr : Review / Q&A session / Exams (see below)

Lab 1: Phillips 265 - Mo, 5:45pm-7:45pm TAs: Ryan Tanner (rjtanner .at. physics.unc.edu) Philip Wulfken (wulfken .at. email.unc.edu) 

Lab 2: Phillips 265 - We, 4:40pm-6:40pm TAs: Ryan Tanner (rjtanner .at. physics.unc.edu)  Andrew Loheac (loheac .at. live.unc.edu)

Office hours: By appointment only. To obtain an appointment, email me directly.   Your subject heading must begin with Phys331.

Website: http://user.physics.unc.edu/~drut/public_html_UNC/physics-331.html Bibliography: - A. Gilat & V. Subramaniam, Numerical Methods for Engineers and Scientists. - Press, Teukolsky, Vetterling, Flannery, Numerical recipes in C (2nd Edition, 1992). - M.L. Boas, Mathematical Methods in the Physical Sciences. Software: MATLAB Midterm 1: Friday, February 12th (in class). Midterm 2: Friday, March 11th (in class). Final Exam: Saturday, April 30th, 12pm, Phillips 265

How to get MATLAB Follow this link: https://software.sites.unc.edu/software/matlab/

Or use our instructions here: http://user.physics.unc.edu/~drut/public_html_UNC/assets/matlabinstructions.pdf

Important dates

Also in the syllabus: Introduction Specific objectives Numerical techniques Programming techniques Grading Attendance policy Homework - There will be a lab this week.

- The homework assignment is posted on our website.

- The due date will always be on the first page of the assignment.

- The homework will always be turned in via Sakai.  - Turn in a PDF file for written part (scan or picture OK)  - More instructions for programming assignment next time

Why numerical techniques?

Why are you here?

Many (in fact most) problems do not have a closed-form analytic solution For example...

Many (in fact most) problems do not have a closed-form analytic solution For example... In classical mechanics Anharmonic motion of a pendulum at large amplitudes The three-body problem The n-body problem for any n > 2 p 1.0 0.5

-1.5 -1.0 -0.5 -0.5 -1.0

0.5

1.0

1.5

x

Many (in fact most) problems do not have a closed-form analytic solution For example... In classical mechanics Anharmonic motion of a pendulum at large amplitudes The three-body problem The n-body problem for any n > 2

p 1.0 0.5 -5

-4

-3

-2

-1

-0.5 -1.0

1

2

x

Many (in fact most) problems do not have a closed-form analytic solution For example... In classical mechanics Anharmonic motion of a pendulum at large amplitudes The three-body problem The n-body problem for any n > 2

Many (in fact most) problems do not have a closed-form analytic solution For example... In electromagnetism

this looks “simple”, but... What if...

... the geometry is complicated (it usually is!)

... the response functions depend on the field (they usually do!)









Many (in fact most) problems do not have a closed-form analytic solution For example... In quantum mechanics Schroedinger equation (single particle) in most potentials The n-body problem for any n > 2

http://phys.org/news/2009-04-quantum-few-body-physics.html

Many (in fact most) problems do not have a closed-form analytic solution For example... In quantum field theory Pretty much any interacting theory in 2D and 3D

http://www.lattice-qcd.org/

Many (in fact most) problems do not have a closed-form analytic solution Some solutions require repetitive tasks For example... Systems of linear equations Root-finding

12.4x + 37y + 238.45z = 20 1.6x + 123y + 19.1z = 1 3.4x + e6 .2y + 7.65z = ⇡

Integration

Image source: wikimedia commons

Many (in fact most) problems do not have a closed-form analytic solution Some solutions require repetitive tasks Sometimes the question we have can be answered without a full analytic solution For example... Thermo- and hydrodynamics Collective phenomena

Lorentzcontracted ions before collision

Quark-gluon plasma (QGP)

Physical results

Predictions

Calculations (Semi-) Analytic

Fully numerical

Theories

What you will get from

this course (hopefully)

Physics problem Hopefully we agree on what problem

we are interested in solving!

Mathematical representation What are the equations that define the problem?

What are the variables? What constitutes a solution?

What is the mathematical language?

Numerical solution on a computer What algorithm are we going to use?

What programming language?

What kind of computer?

Mathematics review Chapter 2 in Gilat and Subramaniam.

If any of this sounds daunting or too foreign,

you may want to:   a) consider taking this course another time;

b) read chapter 2 now as fast as you can.

Calculus Functions Domain, Range Limit Continuity

Intermediate value theorem

Diﬀerentiation

Chain rule, mean-value theorem

Integration

Riemann sum, mean-value theorem

Fundamental theorem of calculus Taylor series

Linear algebra Vectors Definition in 3d and arbitrary dimension Addition and subtraction Multiplication by scalar Transposition Scalar product and orthogonality Linear dependence and independence Triangle inequality

Linear algebra Matrices Definition and relationship with vectors Addition and subtraction Multiplication by scalar Transposition Matrix-matrix multiplication Special matrices: square, diagonal, triangular, identity, symmetric Inverse of a matrix General properties of operations: associativity,

commutativity (or lack thereof),  etc. Determinants and norms

Diﬀerential equations Linear vs. non-linear

Homogeneous vs. inhomogeneous

Order

Analytic solutions

Multivariable calculus Functions of more than one variable

Partial derivatives

Chain rule

Taylor series expansion

Next time: Number representation Sources of error First steps in MATLAB

Reading: Introduction - Chapter 1