This document discusses computational physics and solving physics problems using computers. It outlines the division of labor between physicists and computers, with physicists deriving equations, determining how to instruct computers to solve them, and interpreting results, while computers perform calculations. It describes different types of equations that may be involved, such as algebraic, differential, and those requiring simultaneous solution. Methods for solving equations include analytical solutions, numerical solutions using programming languages, and visualizing solutions with plots.

1. Computational Physics
Introduction
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2. Goals
 Calculate solutions to physics problems
 All physics problems can be formulated
mathematically.
 Many strategies for solving equations.
 Display solutions in a way that helps us
interpret the physics

3. Division of Labor
 Most of the work in solving a problem is still done in
the physicist’s head
 Derive the equations that represent the system of interest
 Understand all the approximations and limitations
(conditions for validity)
 Determine how to instruct the computer to solve the
equations
 The computer does what it is told to do.
 The physicist has to interpret the computer’s output
 Computational solutions are theoretical predictions
(based on the equations). They must eventually be
compared to measurements.

4. equations
 Many types of equations:
 Algebraic
 Trigonometric, logorithmic
 Differential, integral
 Linear, nonlinear
 You may have a set of equations that must
be solved simultaneously
 Matrix manipulations, linear algebra
 Your equations may depend on initial
conditions or boundary conditions

5. Solving equations
 Some special equations have an analytical, or
“closed-form,” solution, which is a certain known
function or combination of functions.
 Most equations (or sets of equations) must be
solved numerically, using a computer. The
resulting solution is approximate, and consists only
of a set of numbers

6. Visualizing results
 In most cases, you will want to make a plot
of the solution, in order to visualize how
certain quantities depend on others.
 This is something that a computer is
especially good at.

7. Strategies
 Programs such as Mathematica and MATLAB can help
you work with equations analytically
 Symbolic manipulation
 Might help you obtain analytical solution
 Most physics problems (that are not highly simplified)
involve equations that must be solved numerically.
 Smooth funtions must be discretized
 Derivatives become differences
 Integrals become sums
 Errors of approximation must be carefully tracked

8. Numerical solutions
 Can be done with Mathematica or MATLAB
 Actual solving strategies are built in
 Need to understand limitations
 Can use a scientific programming language
such as FORTAN or C
 Many basic strategies have already been written
 Use library of routines - customize to your problem
 Solution must be plotted to be useful
 Special graphics programs are available
 Mathematica & MATLAB do this well, too