drawing function phase space and draw the associated vector field

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Elinor Ginzburg 2019-12-22

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评论： Elinor Ginzburg 2019-12-27

hello,

I'm trying to draw the following in phas-space, and to show the vector field, but I'm getting some very strange results. can anyone please explain to me how can I draw this function in phase-space and how to show the vector field?

I'm trying to plot elevation lines where H=constant, and to show the associated vector field.

my attempt:

m=1; R=1;Om=1;g=10;
x= linspace(-100,100,100000);
y=sqrt(m*(R^2)*(Om^2)*(sin(x).^2)-2*(m^2)*g*(R^3)*(cos(x)));
quiver(x,y);

thank you!

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David Goodmanson 2019-12-26

编辑：David Goodmanson 2019-12-27

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Hi Elinor,

there are a couple of things here. The idea is not to calculate p as a function of theta in some manner, as you appear to be doing. Rather, both theta and p are independent variables. The idea is to set up a 2d plane with all possible pairs of values for p and theta, and then calculate H in the theta,p plane. Meshgrid allows you to do that, so meshgrid needs to be done before calculatiing H. And H is simply the expression you provided in the original question, except there is a mistake. The term involving Om^2 should have a + sign..

m = 2;
R = 5;
Om = 0;
g = 9.8;
thetavalues = -3*pi:.1:3*pi;
Lvalues = (-200:.2:200);
[theta,L] = meshgrid(thetavalues,Lvalues);
H = L.^2/(2*m*R^2) + (1/2)*m*R^2*Om^2*sin(theta).^2 - m*g*R*cos(theta);

Using p for the momentum is perfectly acceptable, but since the momentum in this problem is the angular momentum, I changed p to L.

Now you can do a contour plot with theta,L and H. The plot does not really provide enough contour lines, but if you take a look at 'help contour' you can see how to provide more contour lines with one more argument to the contour function. It helps to use 'grid on' just after the contour command, also 'colorbar'.

With Om = 0 you should see stable equilibrium points at theta = -2*pi, 0 and 2*pi, an ordinary pendulum at the low point of its swing. As Om is increased, eventually you should see more equilibrium points appear, at theta values corresponding to the pendulum at the top of its swing.

Elinor Ginzburg 2019-12-27

thank you very much! I understand now how to use it. I'm still new to matlab, so sometimes it's hard to search the documentation for something specific. thank you again for your help!

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