Mathematica -Special applications

Mathematica
- is a computer algebra program, i.e. it performs formula manipulation;
- can do simple algebraic manipulations, integrate, differentiate,
  solve equations, solve differential equations etc.;
- can do numerics e.g. if you need to fit a Voigt function;
- can perform matrix operations, numerical integration, numerical   
  solutions of equations, of differential equations etc.
- can be used to write programs;
- is very useful for a quick overview, for projects where you need
  combinations of algebra, numerics and graphics;
  
- can do everything, but may be not equally well, and with some
  high entrance barrier.

Example 1: Simple differentiation and integration

differentiation

[Graphics:Images/vortrag-pp_gr_1.gif]
[Graphics:Images/vortrag-pp_gr_2.gif]

integration

[Graphics:Images/vortrag-pp_gr_3.gif]
[Graphics:Images/vortrag-pp_gr_4.gif]
[Graphics:Images/vortrag-pp_gr_5.gif]
[Graphics:Images/vortrag-pp_gr_6.gif]

Example 2: Propagation of a scalar wave in a medium:
phase- and group velocity

fictitious refractive index

[Graphics:Images/vortrag-pp_gr_7.gif]

[Graphics:Images/vortrag-pp_gr_8.gif]

dispersion relation

[Graphics:Images/vortrag-pp_gr_9.gif]

[Graphics:Images/vortrag-pp_gr_10.gif]

phase- and group velocity

[Graphics:Images/vortrag-pp_gr_11.gif]

[Graphics:Images/vortrag-pp_gr_12.gif]

assume a gaussian spectral distribution of the wave

[Graphics:Images/vortrag-pp_gr_13.gif]

[Graphics:Images/vortrag-pp_gr_14.gif]

scalar wave

[Graphics:Images/vortrag-pp_gr_15.gif]
[Graphics:Images/vortrag-pp_gr_16.gif]
[Graphics:Images/vortrag-pp_gr_17.gif]

[Graphics:Images/vortrag-pp_gr_18.gif]

time evolution

[Graphics:Images/vortrag-pp_gr_19.gif]

[Graphics:Images/vortrag-pp_gr_120.gif]

Example 3: Web export

make html

[Graphics:Images/vortrag-pp_gr_121.gif]

export

[Graphics:Images/vortrag-pp_gr_122.gif]
[Graphics:Images/vortrag-pp_gr_123.gif]

Example 4: Curve fitting:
determination of the exchange parameters of a magnetic molecule
from high T susceptibility data

[Graphics:Images/vortrag-pp_gr_124.gif]

N=8, s=2/2

[Graphics:Images/vortrag-pp_gr_125.gif]

coupling a=[Graphics:Images/vortrag-pp_gr_126.gif], b=[Graphics:Images/vortrag-pp_gr_127.gif] in Kelvin
H=[Graphics:Images/vortrag-pp_gr_128.gif]

Interaction matrix

[Graphics:Images/vortrag-pp_gr_129.gif]

experimental data for   [Graphics:Images/vortrag-pp_gr_130.gif]

[Graphics:Images/vortrag-pp_gr_131.gif]

the function is known to be a series in 1/T,
try it with various orders

[Graphics:Images/vortrag-pp_gr_132.gif]
[Graphics:Images/vortrag-pp_gr_133.gif]
[Graphics:Images/vortrag-pp_gr_134.gif]
[Graphics:Images/vortrag-pp_gr_135.gif]
[Graphics:Images/vortrag-pp_gr_136.gif]
[Graphics:Images/vortrag-pp_gr_137.gif]
[Graphics:Images/vortrag-pp_gr_138.gif]
[Graphics:Images/vortrag-pp_gr_139.gif]
[Graphics:Images/vortrag-pp_gr_140.gif]
[Graphics:Images/vortrag-pp_gr_141.gif]
[Graphics:Images/vortrag-pp_gr_142.gif]
[Graphics:Images/vortrag-pp_gr_143.gif]
[Graphics:Images/vortrag-pp_gr_144.gif]
[Graphics:Images/vortrag-pp_gr_145.gif]
[Graphics:Images/vortrag-pp_gr_146.gif]
[Graphics:Images/vortrag-pp_gr_147.gif]

plot the data and the approximations

[Graphics:Images/vortrag-pp_gr_148.gif]

[Graphics:Images/vortrag-pp_gr_149.gif]

[Graphics:Images/vortrag-pp_gr_150.gif]

[Graphics:Images/vortrag-pp_gr_151.gif]

[Graphics:Images/vortrag-pp_gr_152.gif]

[Graphics:Images/vortrag-pp_gr_153.gif]

zero-field susceptibility[Graphics:Images/vortrag-pp_gr_154.gif]
[Graphics:Images/vortrag-pp_gr_155.gif]

[Graphics:Images/vortrag-pp_gr_156.gif]

[Graphics:Images/vortrag-pp_gr_157.gif]

[Graphics:Images/vortrag-pp_gr_158.gif]

[Graphics:Images/vortrag-pp_gr_159.gif]

[Graphics:Images/vortrag-pp_gr_160.gif]

[Graphics:Images/vortrag-pp_gr_161.gif]

[Graphics:Images/vortrag-pp_gr_162.gif]

[Graphics:Images/vortrag-pp_gr_163.gif]
[Graphics:Images/vortrag-pp_gr_164.gif]
[Graphics:Images/vortrag-pp_gr_165.gif]
[Graphics:Images/vortrag-pp_gr_166.gif]

higher orders

[Graphics:Images/vortrag-pp_gr_167.gif]

determine coefficients with nonlinear fit

a=20;
b=10;

originally third order

[Graphics:Images/vortrag-pp_gr_168.gif]
[Graphics:Images/vortrag-pp_gr_169.gif]
[Graphics:Images/vortrag-pp_gr_170.gif]
[Graphics:Images/vortrag-pp_gr_171.gif]
[Graphics:Images/vortrag-pp_gr_172.gif]

[Graphics:Images/vortrag-pp_gr_173.gif]

[Graphics:Images/vortrag-pp_gr_174.gif]

[Graphics:Images/vortrag-pp_gr_175.gif]

originally fourth order

[Graphics:Images/vortrag-pp_gr_176.gif]
[Graphics:Images/vortrag-pp_gr_177.gif]
[Graphics:Images/vortrag-pp_gr_178.gif]
[Graphics:Images/vortrag-pp_gr_179.gif]
[Graphics:Images/vortrag-pp_gr_180.gif]

[Graphics:Images/vortrag-pp_gr_181.gif]

[Graphics:Images/vortrag-pp_gr_182.gif]

[Graphics:Images/vortrag-pp_gr_183.gif]

originally fifth order

[Graphics:Images/vortrag-pp_gr_184.gif]
[Graphics:Images/vortrag-pp_gr_185.gif]
[Graphics:Images/vortrag-pp_gr_186.gif]
[Graphics:Images/vortrag-pp_gr_187.gif]
[Graphics:Images/vortrag-pp_gr_188.gif]
[Graphics:Images/vortrag-pp_gr_189.gif]
[Graphics:Images/vortrag-pp_gr_190.gif]

[Graphics:Images/vortrag-pp_gr_191.gif]

[Graphics:Images/vortrag-pp_gr_192.gif]

[Graphics:Images/vortrag-pp_gr_193.gif]

sixt order

[Graphics:Images/vortrag-pp_gr_194.gif]
[Graphics:Images/vortrag-pp_gr_195.gif]
[Graphics:Images/vortrag-pp_gr_196.gif]
[Graphics:Images/vortrag-pp_gr_197.gif]
[Graphics:Images/vortrag-pp_gr_198.gif]
[Graphics:Images/vortrag-pp_gr_199.gif]
[Graphics:Images/vortrag-pp_gr_200.gif]

[Graphics:Images/vortrag-pp_gr_201.gif]

[Graphics:Images/vortrag-pp_gr_202.gif]

[Graphics:Images/vortrag-pp_gr_203.gif]

Example 5: Solving Hamiltons equation of motion:
collisions of Argon clusters

Energies in meV,
Distances in A,
Velocities in A/ps,
Masses in u

Definitions

Number of particles = Teilchenzahl

[Graphics:Images/vortrag-pp_gr_204.gif]

Hamilton-Function and generalized gardients

P and X are defined as arrays, so Mathematica knows what they are

[Graphics:Images/vortrag-pp_gr_205.gif]

Hamilton-Function in P and X,
interaction: Lennard-Jones-(6-12)

[Graphics:Images/vortrag-pp_gr_206.gif]

gradients

[Graphics:Images/vortrag-pp_gr_207.gif]
[Graphics:Images/vortrag-pp_gr_208.gif]
[Graphics:Images/vortrag-pp_gr_209.gif]

dynamics

differential equations

[Graphics:Images/vortrag-pp_gr_210.gif]
[Graphics:Images/vortrag-pp_gr_211.gif]

initial conditions: two clusters of 3 Argon each,
CM1 at {-Rand,0,0}, CM2 at {+Rand,0,1}

[Graphics:Images/vortrag-pp_gr_212.gif]

all equations

[Graphics:Images/vortrag-pp_gr_213.gif]

all momenta and coordinates

[Graphics:Images/vortrag-pp_gr_214.gif]
[Graphics:Images/vortrag-pp_gr_215.gif]
[Graphics:Images/vortrag-pp_gr_216.gif]

solve differential equations numerically

[Graphics:Images/vortrag-pp_gr_217.gif]
[Graphics:Images/vortrag-pp_gr_218.gif]

plot dynamics

[Graphics:Images/vortrag-pp_gr_219.gif]
[Graphics:Images/vortrag-pp_gr_220.gif]
[Graphics:Images/vortrag-pp_gr_221.gif]
[Graphics:Images/vortrag-pp_gr_222.gif]
[Graphics:Images/vortrag-pp_gr_223.gif]

[Graphics:Images/vortrag-pp_gr_224.gif]


Converted by Mathematica      November 26, 2002