Phase Plot
Table of Contents
Phase plot is a vector plot that has phase space (? state space) quantities on the axes and vectors represent their derivatives. Paths connecting the vectors represent the path taken by a system starting or on any point/state of the path.
It is very useful for visualizing Differential Equations (ODE upto 2nd order)
1. Simple Harmonic Oscillator
equation:
\(m u'' + k u' = 0\)
convert this to set of two first order DE:
\(u' = v\) \(v' = -\frac k m u\)
Here the state space is \((u, v)\) and the derivative of this is \((u', v')\)
plotdf takes following arguments:
- the derivative (or derivative vector)
- state variable or state vector
and other optional parameters:
- [parameters, "u=u0, v=v0"] Starting condition
- [sliders, "u=a:b"] Slider for a variable giving range
(Details at Maxima Docs 22.6)
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plotdf([v,-k*u/m], [u,v], [parameters,"m=2,k=2"], [sliders,"m=1:5"], [trajectory_at,6,0])$
2. Damped Pendulum
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plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w], [parameters,"g=9.8,l=0.5,m=0.3,b=0.05"], [trajectory_at,1.05,-9],[tstep,0.01], [a,-10,2], [w,-14,14], [direction,forward], [nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])
3. Plotting Potential Field
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V: 900/((x+1)^2+y^2)^(1/2)-900/((x-1)^2+y^2)^(1/2)$ ploteq(V,[x,-2,2],[y,-2,2],[fieldlines,"blue"])$
