Find the first integral of the following system:
$x_1' = x_2 $
$x_2' = \mu x_1-x_1^3 $ where the first integral is a function such that
$H'(x_1, x_2) = \frac{\partial H(x_1, x_2)}{\partial x1} f_1(x_1, x_2)+ \frac{\partial H(x_1, x_2)}{\partial x2}f_2(x_1, x_2)=0$
In the above example,
$f_1(x_1, x_2) = x_2$
$f_2(x_1, x_2) = \mu x_1-x_1^3 $
In case if it is hard to follow equations I attached the snapshots from the nonlinear book by Sastry.
Could you take a snapshot of the problem in its mathematical form? I could not grasp the equations.
You can use below site by copy/paste into window at that site
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Hi, using MAPLE, the solution of the system above leads for x1(t) to the following expression:
x1 = C2*sqrt(2)*sqrt(mu/(-1+2*mu+C2^2))*JacobiSN(((1/2)*sqrt(2-4*mu)*t+C1)*sqrt(2)*sqrt(mu/(-1+2*mu+C2^2)), C2/sqrt(-1+2*mu))
where C1 and C2 are two constants and where JacobiSN is defined by:
JacobiSN(z, k) = sin(JacobiAM(z, k))
where JacobiAM denotes the Jacobi amplitude function.
A first (energy) integral is
E = x22/2 -\mu x12/2 + x14/4
which is also the (Hamilton) function H from which you can derive the equations of motion.
Divide both sides, then reduce the equation to Pfaffian differential form. Because the system does not have a damping term the Schwarz necessary an sufficient condition for direct integrability is satisfied (equivalently the trace of Jacobi matrix associated with vector field dictated by the RHS of differential equation vanishes). Standard integration procedure of the Pfaffian yields the Hamiltonian just indicated above by Kare Olaussen.
The equation of trajectories associated to this system is (dx_2)/(dx_1)= (\mu x_1-x_1^3)/(x_2) (equation with separable variables) =>
H(x_1,x_2)=Integrate[x_2,x_2]-Integrate[\mu x_1-x_1^3, x_1]
(see Kare Olaussen )
or
H(x_1,x_2)=4*(Integrate[x_2,x_2]-Integrate[\mu x_1-x_1^3, x_1])
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