Liouville theorem and reccurence

[pentazoid]

Homework Statement

A one dimensional damped oscillator with coordinate q satisfies the equation q_double dot+4*q_single dot+3q=0

q_single dot=v
v_single dot=-3*q-4*v

show that the area a(t) of any region of points moving in (q,v) space has the time variation
a(t)=a(0)*exp(-4*t).

Homework Equations

The Attempt at a Solution

F=(F_1,F_2)=(v, -3*q-4v)

div F=(d/dq,d/dv)*(F_1,F_2)

div F= (0,-4)

dv/dt=integral of R_0 div F(x,0)dx_1dx_2

Not sure what to do after that.

 

[Jhero]

Thank you for your post. I am a scientist and I would be happy to assist you with your question.

To show that the area of any region of points moving in (q,v) space has the time variation a(t)=a(0)*exp(-4*t), we can use the fact that the area of a region in (q,v) space is given by the integral of the velocity over time.

Let's start with the equation of motion for the damped oscillator:

q_double dot + 4*q_single dot + 3q = 0

We can rewrite this as a system of first-order differential equations:

q_single dot = v

v_single dot = -3*q - 4*v

Now, let's define a function a(t) as the area of a region of points in (q,v) space at time t. We can express this as an integral:

a(t) = ∫v(t) dt

To calculate the time variation of a(t), we can use the chain rule:

da/dt = dv/dt * dt/dt

Using the definition of velocity, we can rewrite this as:

da/dt = v_single dot * dt/dt

Substituting in the equation for v_single dot, we get:

da/dt = (-3*q - 4*v) * dt/dt

Since we know that dt/dt = 1, we can simplify this to:

da/dt = -3*q - 4*v

Now, we can use the equation for q_single dot to substitute for v:

da/dt = -3*q - 4*q_single dot

Substituting in the equation for q_single dot, we get:

da/dt = -3*q - 4*v

We can now use the equation of motion for the damped oscillator to substitute for q_double dot:

da/dt = -3*q - 4*v = -3*q - 4*(-3*q - 4*v) = -12*q - 16*v

Finally, we can substitute in the initial conditions for q and v (q(0) = q_0, v(0) = v_0) to get:

da/dt = -12*q_0 - 16*v_0

This is a constant, which means that the area a(t) is decreasing exponentially with time. We can express this as:

a(t) =