MATHEMATICA – Computer Simulation

R.C. VermaPhysics DepartmentPunjabi UniversityPatiala – 147 002

PART IX- Computer Simulation

Mechanics

•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.

INTRODUCTION

• 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

What PC can do?

Questions

For practical purposes of PC into physics we need to answer:-

1. How to use PC to improve physics teaching?

2. What other changes will come after we introduce PC to the physics curricula?

3. Could advances of research into physics learning be incorporated into new Curricula?

4. Can the new curricula reflect contemporary physics?

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

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

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.

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 see

what the implication would be. • E.g. What would happen if we change the gravitational

force law a little?

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.

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.

Steps to solving Physics Problem• Identify the input variables:-

like parameters of a physical systeminitial conditions of the`system 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.

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.

One Dimensional Motion

• A spherical body falling in viscous medium

rvmgdt

dvm 6

Clear"Global`" Find analytic solution ndsol DSolve v't g c vt , v0 u , vt, tSimplifyndsolg 9.8; acceleration due to gravity eta 1.0; Viscosity of medium rad 0.2; mass 1.0; radius & mass of ball c 6 eta rad Pi mass N Give initial condition, time interval tmin 0; tmax 2.0; u 1.0;

ndsol

Plot Evaluate vt . ndsol, t, tmin, tmax,AxesLabel "t", "v", PlotRange u, g c,PlotLabel "Drag effect"

Out[18]= vt c t g c t g c u

c

Out[19]= vt c t 1 c t g c u

c

Out[21]= 3.76991

Out[23]= vt 0.265258 3.76991 t 13.5699 9.8 3.76991 t

Out[24]=

0 .5 1 .0 1 .5 2 .0t

1 .0

0 .5

0 .5

1 .0

1 .5

2 .0

2 .5

v

D rag effect

Damped Oscillator

• Equation of motion is

0202

2

xdt

dx

dt

xd

Clear["Global`*"]

(* Find Analytic solution *)

k = 3.0; (* spring constant *)

m = 1.0; (* mass attached to the spring *)

w0 = Sqrt[k/m];

c = 0.5;

damp = c/m;

x0=1.0; v0 = 1.0; (* initial conditions *)

tmin = 0;tmax = 5;

ndsol=DSolve[ {x''[t]+damp*x'[t]+w0^2 x[t]==0,

x[0]==x0, x'[0]==v0}, x[t], t]//Chop//Flatten

1. Cos[1.71391 t] 0.729325 Sin[1.71391 t]{x[t] -> ----------------- + -----------------------} ---------------- ---------------------- 0.25 t 0.25 t E E

(* Plot the solution for a given time interval *)

p1= Plot[ x[t]/.ndsol, {t,tmin, tmax}, AxesLabel->{"t->", "x"}, PlotLabel->"Harmonic Motion"]

1 2 3 4 5t->

Harmonic Motion

-0.75

-0.5

-0.25

0.25

0.5

0.75

1

x

v[t_]= D[x[t]/.ndsol, t]p2 =Plot[ v[t], {t,tmin, tmax},

AxesLabel->{"t->", "v"}, PlotLabel->"velocity",PlotStyle-> Dashing[{0.02} ]]

1. Cos[1.71391 t] 1.89624 Sin[1.71391 t]----------------- - -------------------------------------- --------------------- 0.25 t 0.25 t E E

1 2 3 4 5t->

velocity

-1.5

-1

-0.5

0.5

1

v

a[t_]= D[v[t], t]p3=Plot[ a[t], {t,tmin, tmax},

AxesLabel->{"t->", "a"}, PlotLabel->"acceleration",PlotStyle-> Dashing[{0.05} ]]

-3.5 Cos[1.71391 t] 1.23985 Sin[1.71391 t]------------------- - ---------------------------------------- --------------------- 0.25 t 0.25 t E E

1 2 3 4 5t->

acceleration

-3

-2

-1

1

2

a

Show[p1, p2, p3]

1 2 3 4 5t->

Harmonic Motion

-3

-2

-1

1

2

x

ParametricPlot[ {x[t]/.ndsol, v[t]} , {t,tmin, tmax}

, AxesLabel->{"x", "v"}, PlotLabel->"phase_space_trajectory"]

-0.75 -0.5 -0.25 0.25 0.5 0.75 1x

phase_space_trajectory

-1.5

-1

-0.5

0.5

1

v

Thank you!