PHYS385
Numerical Methods I, How to Solve Problem on a computer
Instructor: Prof. Varley Tuesdays and Thursdays 7:00-8:15PM
This course is taught in an electronic classroom room 1000, Lab B North Bldg (Macintosh) using Mathematica, as the programming and graphing language.
Textbook: Computational Physics 2nd edition by Nicholas Giordano and H. Nakanishi. (Prentice Hall, New Jersey). Chapters 1-4 and appendices A, B, D, and E are studied in PHYS385/685. The remainder of the book is studied in PHYS 485/695.
Introductory Comments
Much of physics is described by differential equations and it is therefore
important that the student understand how to solve differential equations.
Only a few kinds of differential equations can be solved exactly for simple
models, and for real world problems, approximation methods or numerical
methods must be used. It used to be that physics was broadly divided into
theory and experiment but since about 1945 a new area called "computational
physics" or numerical simulations has become important.
Numerical
simulations uses computational tools on computers to solve physics problems
in a manner that borrows from both theory and experiment. The actual theory
used on the computer is either classical or quantum physics but the problem
is explored in the manner of an experimentalist. Specifically a parameter is
varied in the problem and the effect of that variation is observed in the
computer output instead of an experiment. The computer simulation is a sort
of "computer experiment". The methods of solving differential equations
on
a computer start with the Euler method and progress through Runge-Kutta.
The
course will start with solving simple radioactive decay problems and heating/cooling
problems so as to build confidence in the numerical methods of solution. Projectile
problems with air resistance are studied and this is
important as air resistance plays an important role in most real situations.
Theoretical courses often neglect air resistance as it make the calculation
too difficult but computational methods are very easy.
The mass on a spring
problem is studied not just when the spring obeys the Hook's linear force
law, but also when non-linear effects are important. Again numerical methods
treats these nonlinear problems quite easily while the corresponding
theoretical methods can be quite complex. Various realistic models of the
solar system will be studied with computational methods and this is to be
contrasted with the theoretical methods which have difficulty much beyond the
two body problem. Finally, the various aspects of chaos or random that
appear in nonlinear problems (like the "Butterfly effect") will
be
examined.
Quite often in physics various integrals appear as part of the
solution and it is important to be able to "perform" these integrals. Mathematica can do a huge number of integrals symbolically
and the student will learn
how to use this feature for use in theoretical physics courses and research.
However, many integrals that appear in physics can only be done numerically
and various methods of efficiently, performing integrals numerically are
discussed including Newton-Cotes, trapezoid rule, and Gauss method. Also,
discussed are various methods of doing numerical derivatives as appear in the
laboratory etc. Additionally various techniques for fitting experimental
data to a function will be discussed including least square fitting using not
just linear functions but polynomial, exponential, Sine functions etc.
Additionally, root finding of polynomial and transcendental equations is
useful and will be taught using for example Newton-Raphson and secant
methods
Pre-req: PHYS121 or 120 and two semesters of calculus. MATH 254 (ordinary differential equations) is NOT a pre-req. MATH 254 is not necessary for the student since the numerical techniques for solving differential equations taught in PHYS 385/685 are quite different from the techniques taught in MATH 254.
GRADING: There will be two midterm exams and a final exam. The highest of
the two midterm exams counts 30% toward the final course grade and the final
exam counts 40% toward the course grade.
Important Note: Work submitted by
students on exams is required to be their own work and copying from others
on
exams (including take home exams) is consider plagiarism and subject to
Hunter College rules of penalties of paralogism. The exams will begin in
class and then continue on as take home exams. Homeworks assignments
(workshops) are an important part of the course as there are eighteen
workshop modules posted on the web (roughly one workshop per two lectures).
There will be time set aside in class for students to work on the workshop.
Students are encouraged to work with a partner although each student is
expected to turn in their own workshop done in their own way (that is,
students may NOT copy workshops and hand them in). The workshops count 30%
toward the course grade. Attendance in class is required and is part of the
workshop grade.
Hunter College regards acts of academic
dishonesty (e.g., plagiarism, cheating on examinations, obtaining unfair
advantage, and falsification of records and official documents) as serious
offenses against the values of intellectual honesty.
The college is committed to enforcing the CUNY Policy on Academic
Integrity and will pursue cases of academic dishonesty according to the
Hunter College Academic Integrity Procedures.