Question

Find the implicit equations of the trajectories of the following conservative systems. Next find their critical points (if any) and classify them.

x" + x + x = 0

Answer 1

The trajectories of the system are concentric circles centered at the critical point (0,0).

The given second-order differential equation is x′′ +x+x=0, which can be written in the form x′′ +2x=0. This is a simple harmonic oscillator equation with a natural frequency of ω= √2

​To find the implicit equations of the trajectories, we can first write the system as a set of first-order differential equations. Let v=x′ , then the system becomes:

x′ =v

v′=−x

​Now, we have a system of first-order differential equations:

dx/dt= =v

​dv/ dt =−x

This system can be written in vector form as u′ =f(u), where

u=
`\left[\begin{array}{ccc}x\\v\end{array}\right]` and f(u)=
`\left[\begin{array}{ccc}v\\-x\end{array}\right]`

Now, to find the critical points, we set f(u) equal to the zero vector:

v=0

−x=0

​​This gives us two critical points:

(x,v)=(0,0).

To classify the critical points, we can consider the linearization of the system around each critical point. The Jacobian matrix J is given by:

J=
`\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]`

Evaluating this matrix at the critical point (0,0), we get:

J(0,0)=
`\left[\begin{array}{ccc}0&1\\-1&0\end{array}\right]`

The eigenvalues of this matrix are λ=±i, which indicates that the critical point is a center. Therefore, the trajectories of the system are concentric circles centered at the critical point (0,0).