Oscillator Equation - Energy Conservation

[Paddyod1509]

Given the Oscillator equation:

$\frac{d2s}{dt2}$ + $\omega$²s = 0

Show that the energy:

E=1/2($\frac{ds}{dt}$)² + 1/2$\omega$²s²

is conserved.

Any help at all appreciated! Thankyou

 

[Paddyod1509]

I think i have it. By assuming a solution of the form

s(t)=Acos(wt)

and showing that the energy = 1/2(A^2)(w^2) is constant, this proves conservation

 

[cosmic dust]

There is a more direct approach: just take the derivative w.r.t. time of the energy exression and then use the differential equation to replace the second derivative. If the derivative of energy is zero then the energy is conserved...

 

[Paddyod1509]

hi cosmic dust, thanks for your reply!

so by taking the derivative wrt time of the energy expression, i get

dE/dt = s' s'' + s'(ω^2s^2)

which is just s' times the given oscillator equation, which is zero, so:

dE/dt = s'(0)=0

So basically, by showing dE/dt = 0, i have shown that energy is conserved?

 

[K^2]

So basically, by showing dE/dt = 0, i have shown that energy is conserved?

Precisely.

One of your lines has a typo, though. It's s'(ω²s) not s².

 

[Paddyod1509]

Thankyou sir, much appreciated