Oscillator Equation - Energy Conservation

AI Thread Summary
The discussion centers on demonstrating the conservation of energy in the oscillator equation d²s/dt² + ω²s = 0. By assuming a solution of the form s(t) = Acos(ωt), it is shown that the energy E = 1/2(ds/dt)² + 1/2ω²s² remains constant. A more direct method involves taking the time derivative of the energy expression, resulting in dE/dt = s' s'' + s'(ω²s), which simplifies to zero using the oscillator equation. This confirms that energy is conserved, as dE/dt = 0. The conversation concludes with a clarification on a minor typo in the equations presented.
Paddyod1509
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Given the Oscillator equation:

\frac{d2s}{dt2} + \omega2s = 0

Show that the energy:

E=1/2(\frac{ds}{dt})2 + 1/2\omega2s2

is conserved.

Any help at all appreciated! Thankyou
 
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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
 
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...
 
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?
 
Paddyod1509 said:
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.
 
Thankyou sir, much appreciated
 
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