You are Here: Home >< Maths

# Phase planes of Conservative systems (2nd order ODEs) watch

1. I'm unsure whether my phase plane diagram is correct or not for this question. Could someone verify?

Question:
Spoiler:
Show

Phase diagram for pendulum eq. from lec. notes:

Spoiler:
Show

My attempt:

Determine potential energy (up to a const.)

Total energy eq. is hence for some (this is the energy level) and

Equilibrium points given by thus

which is for and for so the former is a local min point, and the latter are the two max points.

Sketching against , and against below it, gives:

Spoiler:
Show

2. From my understanding you're just drawing the family of curves for different values of h. If we fix w, then you have the right idea for 0<h<(1/24)w^2(6)(12-(6)) = U(sqrt(6)), i.e ellipses at the centre.

But for h>U(sqrt(6)), your reasoning is a bit off. For x-> +- infinity from the U(x) graph you can see that U-> -infinity, so as you have (1/2) y^2 + U = h this implies (1/2) y^2 = h - U > 0 (clear as we took h greater than max value of U). So as x-> +- infinity, U-> -infinity, so y^2 -> infinity. Of course this makes sense as we expect energy to be conserved so as U decreases KE increases.

For h<0, there are some more considerations to make, but you can try reasoning like above. Also for 0<h<U(sqrt(6)), you should think about what happens when x as it gets bigger.

You can try plotting these graphs for varying h.
Hope that helps.
3. (Original post by MagneticFlux)
From my understanding you're just drawing the family of curves for different values of h. If we fix w, then you have the right idea for 0<h<(1/24)w^2(6)(12-(6)) = U(sqrt(6)), i.e ellipses at the centre.

But for h>U(sqrt(6)), your reasoning is a bit off. For x-> +- infinity from the U(x) graph you can see that U-> -infinity, so as you have (1/2) y^2 + U = h this implies (1/2) y^2 = h - U > 0 (clear as we took h greater than max value of U). So as x-> +- infinity, U-> -infinity, so y^2 -> infinity. Of course this makes sense as we expect energy to be conserved so as U decreases KE increases.

For h<0, there are some more considerations to make, but you can try reasoning like above. Also for 0<h<U(sqrt(6)), you should think about what happens when x as it gets bigger.

You can try plotting these graphs for varying h.
Hope that helps.
Thanks, that helped. Does this look about right now?

Spoiler:
Show

4. (Original post by RDKGames)
Thanks, that helped. Does this look about right now?

Spoiler:
Show

Yeah that's what I had in mind, nicely done.

TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: February 7, 2018
Today on TSR

### Life lessons uni will teach you

Questionable decision-making and constant surprises

### University open days

1. Loughborough University
Fri, 21 Sep '18
2. University of Cambridge
Fri, 21 Sep '18
3. Richmond, The American International University in London
Fri, 21 Sep '18
Poll
Useful resources

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams