How should I find the equilibrium points and the general equation?

[Math100]

Homework Statement

Find the equilibrium points and the general equation for the phase paths of $\ddot{x}+cos(x)=0$. Obtain the equation of the phase path joining two adjacent saddles.

Relevant Equations

For the general equation $\ddot{x}=f(x, \dot{x})$, equilibrium points lie on the $x$ axis, and are given by all solutions of $f(x, 0)=0$, and the phase paths in the plane $(x, y) (y=\dot{x})$ are given by all solutions of the first-order equation $\frac{dy}{dx}=\frac{f(x, y)}{y}$.

Consider the differential equation $\ddot{x}+cos(x)=0$.
Note that $\ddot{x}=f(x, \dot{x})$, so we have $f(x, y)=-cos(x)$.
Then $f(x, 0)=-cos(x)=0$.
This gives $x=n\pi-\frac{\pi}{2}$ for some $n\in\mathbb{Z}$.
Since the differential equation for the phase paths is given by $\frac{dy}{dx}=-\frac{cos(x)}{y}$,
it follows that $y dy=-cos(x)dx\implies \int y dy=-\int cos(x)dx\implies \frac{y^2}{2}=-sin(x)+C$.
Thus, $y^2=-2sin(x)+C\implies y=\pm\sqrt{C-2sin(x)}$ where $C$ is an arbitrary constant.

Above is my work for this problem. However, I've only found the general equation/solution for the phase paths of $\ddot{x}+cos(x)=0$ but how should I find the equilibrium points based on the equation I've found above in my work $x=n\pi-\frac{\pi}{2}$ for some $n\in\mathbb{Z}$? Also, how should I obtain the equation of the phase path joining two adjacent saddles from here?

 

[pasmith]

The fixed points are at $(\dot x, \ddot x) = (0,0)$. You have correctly found these to be at $(x, \dot x) = ((n + \frac12)\pi,0), n \in \mathbb{Z}$. But which are saddles? You need to find the eigenvalues of the Jacobian at these fixed points to determine that.

Then you can find the value of $C$ at a saddle point, which will give you the equation of the phase path(s) connecting adjacent saddles.

 

[Math100]

The fixed points are at $(\dot x, \ddot x) = (0,0)$. You have correctly found these to be at $(x, \dot x) = ((n + \frac12)\pi,0), n \in \mathbb{Z}$. But which are saddles? You need to find the eigenvalues of the Jacobian at these fixed points to determine that.

Then you can find the value of $C$ at a saddle point, which will give you the equation of the phase path(s) connecting adjacent saddles.

Why is it $(x, \dot{x})=((n+\frac{1}{2})\pi, 0)$ for some $n\in\mathbb{Z}$ instead of $(x, \dot{x})=((n-\frac{1}{2})\pi, 0)$ for some $n\in\mathbb{Z}$? Also, what's the Jacobian in this problem and how to find the eigenvalues of this Jacobian in order to find those saddles?