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File:Anharmonic oscillators solutions.svg

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Description
English: solution to several anharmonic oscillator problems
Date
Source Own work
Author Jkrieger
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Sourcecode edit

Note: This script need the plot2svg() function package by Juerg Schwizer!

 
The SVG code is valid.
 
This diagram was created with MATLAB.

Source code

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MATLAB code

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% three function files 
function [ dy ] = harmonic( t,y,m,k,l,c )
%UNTITLED3 Summary of this function goes here
%   Detailed explanation goes here
dy=zeros(2,1);
dy(1)=y(2);
dy(2)=-(c*y(2)+k*y(1))/m;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [ dy ] = anharmonicA( t,y,m,k,l,c )
%UNTITLED3 Summary of this function goes here
%   Detailed explanation goes here
dy=zeros(2,1);
dy(1)=y(2);
dy(2)=-(c*y(2)+k*y(1)+l*y(1)^2)/m;
end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

function [ dy ] = anharmonicB( t,y,m,k,l,c )
%UNTITLED3 Summary of this function goes here
%   Detailed explanation goes here
dy=zeros(2,1);
dy(1)=y(2);
dy(2)=-(c*y(2)+k*y(1)+l*y(1)^3)/m;
end



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% main script
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
addpath('plot2svg');

m=1;
k=1;
l=0.2;
c=0;
x0=0.2;
x01=2.4;
x02=0;
p02=2;
p0=0;

PSX0range=0:(0.4):(2.4);

PSRange=[-4.5 4.5];
 
 
 
dy=@(t,y) anharmonicA(t,y,m,k,l,c);
dyB=@(t,y) anharmonicB(t,y,m,k,l,c);
dyh=@(t,y) harmonic(t,y,m,k,l,c);
options = odeset('RelTol',1e-4,'AbsTol',[1e-4 1e-4]);
 
 
h=figure(1);
subplot(3,2,1);
[T,Y] = ode45(dy,[0 40],[x0, p0],options);
[Th,Yh] = ode45(dyh,[0 40],[x0, p0],options);
[TB,YB] = ode45(dyB,[0 40],[x0, p0],options);
plot(Th,Yh(:,1), 'b-', T,Y(:,1),'r-', TB,YB(:,1),'g-');
ylim([-1.5 1.5]*x0);
xlabel('Zeit t');
ylabel('Ort x(t)');
title(['\bf Trajektorie für kleine Anfangsauslenkung (x_0=' num2str(x0) ', p_0=0)']);
 
subplot(3,2,2);
for x0=PSX0range
    [Th,Yh] = ode45(dyh,[0 10],[x0, p0],options);
    plot(Yh(:,1),Yh(:,2), 'b-');
    if x0==PSX0range(1)
        hold on
    end
end
hold off
xlim(PSRange);
ylim(PSRange);
daspect([1 1 1]);
xlabel('position x');
ylabel('impulse p');
title(['\bf Phasenraum-Plot, harmonischer Oszillator']);
 
 
subplot(3,2,3);
[T,Y] = ode45(dy,[0 40],[x01, p0],options);
[Th,Yh] = ode45(dyh,[0 40],[x01, p0],options);
[TB,YB] = ode45(dyB,[0 40],[x01, p0],options);
plot(Th,Yh(:,1), 'b-', T,Y(:,1),'r-', TB,YB(:,1),'g-');
xlabel('Zeit t');
ylabel('Ort x(t)');
title(['\bf Trajektorie für große Anfangsauslenkung (x_0=' num2str(x01) ', p_0=0)']);
ylim([-2 2]*x01);
subplot(3,2,4);
for x0=PSX0range
    [T,Y] = ode45(dy,[0 10],[x0, p0],options);
    plot(Y(:,1),Y(:,2), 'r-');
    if x0==PSX0range(1)
        hold on
    end
end
hold off
xlim(PSRange);
ylim(PSRange);
daspect([1 1 1]);
xlabel('Ort x');
ylabel('Impuls p');
title(['\bf Phasenraum-Plot, asymmetrisches Kraftgesetz']);
 
 
subplot(3,2,5);
[T,Y] = ode45(dy,[0 40],[x02, p02],options);
[Th,Yh] = ode45(dyh,[0 40],[x02, p02],options);
[TB,YB] = ode45(dyB,[0 40],[x02, p02],options);
plot(Th,Yh(:,1), 'b-', T,Y(:,1),'r-', TB,YB(:,1),'g-', [0 40], [2 2], 'b:', [0 40], -[2 2], 'b:');
xlabel('Zeit t');
ylabel('Ort x(t)');
title(['\bf Trajektorie bei gleicher Energie (x_0=' num2str(x02) ', p_0=' num2str(p02) ')' ]);
ylim([-2.5 2.5]);
subplot(3,2,6);
for x0=PSX0range
    [TB,YB] = ode45(dyB,[0 10],[x0, p0],options);
    plot(YB(:,1),YB(:,2), 'g-');
    if x0==PSX0range(1)
        hold on
    end
end
hold off
xlim(PSRange);
ylim(PSRange);
daspect([1 1 1]);
xlabel('Ort x');
ylabel('Impuls p');
title(['\bf Phasenraum-Plot, symmetrisches Kraftgesetz']);

% uses the plot2svg tool from MatlabCentral by Juerg Schwizer 
plot2svg('anharmonic_oscillators_solutions.svg', h)

Licensing edit

I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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Date/TimeThumbnailDimensionsUserComment
current05:57, 14 August 2013Thumbnail for version as of 05:57, 14 August 2013512 × 470 (138 KB)Jkrieger (talk | contribs)x_0->x(0), p_0->p(0), added dashed lines
15:27, 1 June 2013Thumbnail for version as of 15:27, 1 June 2013512 × 470 (153 KB)Jkrieger (talk | contribs)Category:Theoretical physicists Category:Created with MATLAB

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