Find the energy function and show that energy decreases with time

Question

Given the system of a damped pendulum: $$\dot{x}=y\\\dot{y}=-(x+2x^3)-0.1y$$

Find the total mechanical energy function $E(x,y)$ of the undamped pendulum and show that the energy decreases with time.

I found that $$E(x,y)=\frac{y^2}{2}+\frac{x^2}{2}+\frac{x^4}{2}$$ But I don't know how to show that energy is decreasing with time.

Answer 1

We evaluate the total derivative of $t\to E(x(t),y(t))$: $$\frac{d(E(x(t),y(t))}{dt}=y\dot{y}+x\dot{x}+2x^3\dot{x} =y(-(x+2x^3)-0.1y)+(x+2x^3)y=-0.1y^2\leq 0$$ Hence energy is decreasing with time along the solutions of the system.