MATHEMATICA

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MATHEMATICA Computer Simulation R.C.
Verma Physics Department Punjabi
University Patiala 147 002 PART IX- Computer
Simulation Mechanics
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INTRODUCTION

• Traditionally physics teaching comprises of
  theory lectures based on analytical techniques
  and conventional laboratory experiments.
• Despite the importance of computational physics,
  it has been largely neglected in the
  conventional physics curricula.
• Now with the availability of personal computers,
  it has become possible to introduce this
  important branch in the physics curricula.

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What PC can do?

• PC offers new opportunities for innovative
  learning.
• It provides highly interactive, individual and
  creative learning.
• It can help to approach wide variety of problems
  and phenomena than is possible with only analytic
  tools.
• It can also be used to develop physical intuition
  and ability to estimate physical quantities
  involved in a phenomena.
• NMEICT (MHRD, Govt. of India)- Rs. 4,612 crores
  Mission

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Questions

• For practical purposes of PC into physics we
  need to answer-
• How to use PC to improve physics teaching?
• What other changes will come after we introduce
  PC to the physics curricula?
• Could advances of research into physics learning
  be incorporated into new Curricula?
• Can the new curricula reflect contemporary
  physics?

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OBJECTIVES OF PHYSICS TEACHING

• i) Number awareness
• ii) Experimental skills
• iii) Analytic skills
• iv) Scales and estimations
• v) Approximations skills
• vi) Numerical skills
• vii) Intuition large problem skills

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Applications of Computer for Physics?

• Problem Solving
• Demonstrations and Tutorials (CAI)
• Data analysis using Spreadsheets
• Simulation of Physics Problems
• Graphics and Animation
• Magnification of Instruction

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Problem Solving

• PC can be used easily and interactively through
  a variety of high-level languages,
• They offer numerical power sufficient for even
  initiating research-level problems.
• Many numerical programming languages are already
  with us BASIC, FORTRAN, and C
• Recently, Symbolic Computational languages
  Mathematica, MatLab, MathCad, Macsyma
• capable of dealing with algebra, differential and
  integral calculus, and powerful graphics tools.
• This obviously enhances the scope of physics
  problems to be handled on a PC.

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Simulation of Physics Problems

• The corner stone of computing is building a model
  of an idea through simulation.
• It can deliver real time sequence on the screen.
• We can simulate real world phenomena that are
  prevented from studying in the laboratory due to
  constraints of time, expense, danger and
  feasibility.
• E.g. Planetary Motion, Nuclear Reactor, Interior
  of Sun
• We can try models that don't occur in real world
  to seewhat the implication would be.
• E.g. What would happen if we change the
  gravitational force law a little?

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Present Status in Physics Curricula

• Computational physics has largely been neglected
  in the standard physics curricula.
• Main factors 1. Lack of computing hardware 2.
  Lack of teaching-material besides 3. Lack of
  trained human resource.
• Situation is slowly improving.

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COMPUTER SIMULATION OF PROBLEMS (Methodology)

• Physics ? Algorithm ? Program ? Results
• Computer Hardware and Software
• Numerical analysis
• Development of algorithms for problems
• Developments of programs for simulation
• Results and Error analysis.

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Steps to solving Physics Problem

• Identify the input variables- like parameters
  of a physical system initial conditions of
  thesystem and time interval step size of
  time evolution.
• Identify the output variables-
  solution of the problem.
• Construct the equations to connect the
  input variables to the output variables.
• Re-express the equations using numerical
  techniques.
• Write algorithm/flowchart to solve the problem.
• Develop Programs (I/O, common arithmetic
  operations and logical structures
  Sequential, Repetitive and Selective).
• Execute the program on a computer.

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Performing Computer Experiments

• Run computer experiments to study effects of
• change of step size used in discretization of
  continuous independent variable
• change of initial conditions of the physical
  system
• change of physical parameters of the system.
• changes due to errors,
• stability and limitations of the numerical tools.

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One Dimensional Motion

• A spherical body falling in viscous medium

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Damped Oscillator

• Equation of motion is

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Clear"Global" ( Find Analytic solution
) k 3.0 ( spring constant ) m 1.0 (
mass attached to the spring ) w0 Sqrtk/m c
0.5 damp c/m x01.0 v0 1.0 ( initial
conditions ) tmin 0tmax 5 ndsolDSolve
x''tdampx'tw02 xt0,
x0x0, x'0v0, xt, t//Chop//Flatten
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( Plot the solution for a given time interval
)p1 Plot xt/.ndsol, t,tmin, tmax,
AxesLabel-gt"t-gt", "x", PlotLabel-gt"Harmonic
Motion"
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vt_ Dxt/.ndsol, tp2 Plot vt, t,tmin,
tmax, AxesLabel-gt"t-gt", "v",
PlotLabel-gt"velocity", PlotStyle-gt
Dashing0.02
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at_ Dvt, tp3Plot at, t,tmin, tmax,
AxesLabel-gt"t-gt", "a", PlotLabel-gt"accelerat
ion", PlotStyle-gt Dashing0.05
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Showp1, p2, p3
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ParametricPlot xt/.ndsol, vt ,
t,tmin, tmax , AxesLabel-gt"x", "v",
PlotLabel-gt"phase_space_trajectory"
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Thank you!