# Thread: How do I write something as a first order system, or vector field, on the phase plane

1. ## How do I write something as a first order system, or vector field, on the phase plane

I have been asked to write d2s/dt^2 = −s as first order system, or vector field, on the phase plane.

and then compute its potential, I have no idea how to do either.

I have looked it up, I am currently on this website Pauls Online Notes : Differential Equations - Systems of Differential Equations and still don't understand. It's not in my course notes either.

Please help if you have any idea, I have an exam in a couple of week and NEED to know this.

2. Originally Posted by supaman5
I have been asked to write d2s/dt^2 = −s as first order system, or vector field, on the phase plane.

and then compute its potential, I have no idea how to do either.

I have looked it up, I am currently on this website Pauls Online Notes : Differential Equations - Systems of Differential Equations and still don't understand. It's not in my course notes either.

Please help if you have any idea, I have an exam in a couple of week and NEED to know this.
You need to make the substitution

$\displaystyle x=\frac{ds}{dt} \implies \frac{dx}{dt}=\frac{d^2s}{dt^2}$

Now using the 2nd equation and the ODE we get that

$\displaystyle \frac{dx}{dt}=\frac{d^2}{dt^2}=-s$

So now we have the first order linear system of ODE's

$\displaystyle \dot{s}=x$

$\displaystyle \dot{x}=-s$

$\displaystyle \mathbf{v}(s,x)=<x,-s>$

I am not sure what you want for the 2nd part. Are you looking for a scalar function

$\displaystyle \phi$

such that

$\displaystyle \nabla \phi =\mathbf{v}$

3. I think I need to find the potential energy so that the equation 0.5mv^2+V=E(total energy)

and thanks so much for the first part

4. ## Re: How do I write something as a first order system, or vector field, on the phase p

would there not be a constant of integration

5. ## Re: How do I write something as a first order system, or vector field, on the phase p

would it not be $\displaystyle \frac{1}{2} \dot{s}^2+\frac{s^2}{2}= constant=E$ for some E

to compute its potential
compare with $\displaystyle \ddot{s}=-\nabla(s)$
so
$\displaystyle \nabla(s)=s$
so
$\displaystyle V(s)=\frac{s^2}{2}+K$
then choose K to be equal to zero