Computational Physics
Introduction
3/30/11

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

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.

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

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

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.

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

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