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

**supaman5** · May 4th 2011, 08:13 AM

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.

**TheEmptySet** · May 4th 2011, 08:32 AM

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

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

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

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

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

$\dot{s}=x$

$\dot{x}=-s$

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

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

$\phi$

such that

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

**supaman5** · May 4th 2011, 09:17 AM

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

**FGT12** · April 26th 2012, 06:26 AM

would there not be a constant of integration

**FGT12** · April 26th 2012, 06:49 AM

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

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

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

LinkBack

Thread Tools

Display

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

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

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

Similar Math Help Forum Discussions

Nonlinear terms - Phase plane

Eigenvectors for the Phase Plane of a Two-Dimensional Linear System

Phase plane problem

'write the following as a system of higher-order equations in standard form'

Linear System: Phase Planes

Search Tags