@Preben Alsholm 

That's right, cos(q) is cos(q(t)) and the q(t) is the solution of the ODE above.

Because I don't know how to integral with the variable of t so I used q.

@Rouben Rostamian  

1.This H is called hamiltonian function if omega is a constant and H is called adiabatic invariants when omega is growing with respect to t very slowly. That's a right definition, you can google it.

2.That's the reason I give three pictures of the area with respect to t.They are in the first photo at the time t<t*,t=t*and t>t* .

3.I just need the area to calculate I so I can get the picture of I with respect to t. And I is a piesewise function 

4.As you can see in the second photo, t* is the critical value of deciding whether the phase diagram of p-q is closed or opened. t<t* it's open , t=t* it's closed at the points q=+-pi and t>t* it's closed. By given H and omega I can get a t* so I can draw the phase diagram.And the t* also the critical value of I change into 2I because the when the curve is closed the area is 2 time as before.

5. A..that's what I don't know, thanks for telling me that.

I am not familiar with the definition 'autonomous',I just know the equation is an ODE, nothing more . And I an a beginner to MAPLE, so this question is quite difficult to me .

If you still confuse about something I replied this time please let me know, thanks.

@Kitonum 

I tried to solve that but there is an error. And I don't know how to give value to H which is a constant and satisfied the equation in the paper.

This is another code which is different from the reply hw2_numerical_1.mw

@Kitonum 

good idea! 

But I met an error, this is my code.

restart;

w := 1+(1/10)*t;

sol:=dsolve({diff(q(t),t$2)+w*sin(q(t))=0, p(t)=diff(q(t),t),diff(I(q(t),q(t))=(p(t))/(2 pi), q(0)=1, D(q)(0)=2}, {q(t),p(t),I(t)}, numeric);

Error, unable to match delimiters

sol:=dsolve({diff(q(t),t$2)+w*sin(q(t))=0, p(t)=diff(q(t),t),diff(I(q(t),q(t))=(p(t))/(2 Pi), q(0)=1, D(q)(0)=2}, {q(t),p(t),I(t)}, numeric);

plots[odeplot](sol, [[t, (I)(t)]], t = -10 .. 10, color = [red, blue], thickness = 1, legend = ['E(t)', '(I)(t)']);

@Kitonum 

This is the link of my question.

http://www.mapleprimes.com/questions/206645-How-To--Numerically--Dsolve-It-

I am a beginner and don't know how to integral function.It's quite difficult for me

By the way , how do you become so great at maple?I looked some books in library but found out they are not friendly to a beginner, could you tell me how to learn it by myself?

@Kitonum 

Thanks very much!

@vv @Kitonum 

My tutor told me he want the plot made in numerical method beause there will be higher order ODE waiting for me and he told me the solution can't be solved unless use the numerical method.

So how could this question be solved in a numerical method? Could you give me a code?

@Markiyan Hirnyk 

I am a greenhand , don't know what is listprocedure..

And the matrix is the dsn

@tomleslie @Rouben Rostamian  

Thank you guys, I really appreciate it : )

Another question, how did you study maple ? read a book or learn at google? Could you recommend me some materials?

@vv @Kitonum

Thanks so much, both of you !

I really appreaciate it , you guys saved me , best wishes  : )

@Kitonum 

Thanks for your help but it's not what I want . Could you please help me write another code ? 

I want to get a plot of E and I with respect to t.

E=0.5*(diff(x(t),t)^2+w^2*x(t)^2), w=1+t/100.

I=E/w.

Sorry for wasting your time because of my fault that I didn't illustrate my question clearly.

If you can help me I would really appreaciate it : )  

@Carl Love 

The equation is the second line in the paper : diff(x(t),t$2)+w^2x(t)=0

w=1+t/100

I let y(t) to be diff(x(t),t).

Even if I copy the code to here you won't understand it because I don't know how to compose it.