# 2nd order DEQ: conserved quantity pt 2

1. Mar 13, 2007

### fusi0n

1. The problem statement, all variables and given/known data

Consider y'' = - sin(y)

find a conserved quantity for this equation

2. Relevant equations

This looks an awful lot like a simplified version of a nonlinear pendulum equation

3. The attempt at a solution

For a conserved quantity I guessed: E = -cos(y) + y' because we need a y'' upon differentiating E and also we will need to cancel out the -sin(y)

=> dE/dt = sin(y)*y' + y''

=> y'' = (dE/dt) - sin(y)*y' = -sin(y)

=> dE/dt = sin(y)*y' - sin(y) = sin(y)*(y'-1) and does not equal zero so E is not conserved.

2. Mar 13, 2007

### Dick

Mess around with combinations of (y')^2 and cos(y). Why did I choose those two?

3. Mar 13, 2007

### fusi0n

Thanks. E = (1/2)y'^2 -cos(y) works. The y'^2 allows for an additional y' term that can cancel later!

4. Mar 13, 2007

### arildno

Multiply your original equation with y' and scrutinize the slip of paper you have written it on.