1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Calculating the energy of a harmonic oscillator

  1. Aug 2, 2011 #1
    the general solution is given by x(t) = Acos(ωt) + Bsin(ωt). Express the total energy in terms of A and B and notice how it is independent of time.

    my book derives a formula earlier which says [tex]\frac{\partial{S_{cl}}}{\partial{t_f}} = -E[/tex] where [tex]S_{cl}[/tex] is the classical path defined by [tex]S_{cl} = \int L dt [/tex] where L is the lagrangian.

    i know that the lagrangian for a harmonic oscillator is L = [tex]\frac{1}{2}m\dot{x}^2-\frac{1}{2}kx^2[/tex]

    so i tried [tex]\frac{\partial{S}}{\partial{t}}[/tex] = [tex] \int (\frac{\partial{L}}{\partial{x}} \dot{x} dt + \frac{\partial{L}}{\partial{\dot{x}}}\frac{d\dot{x}}{dt} dt)[/tex]

    and i have [tex]\dot{x} = -A\omega\sin(\omega t)+B\omega\cos(\omega t) [/tex] as well as [tex]\ddot{x} = -A\omega^2\cos(\omega t)-B\omega^2\sin(\omega t) [/tex]

    so after substituting and taking the antiderivative, i get something very messy:
    [tex]\frac{1}{2}(ma^2\omega^2 + kB^2\omega^3)(t - \frac{1}{2\omega}\sin{2\omega t})+\frac{1}{2}(mB^2\omega^2 + kA^2\omega^3)(t+\frac{1}{2\omega}\sin{2\omega t}) - \frac{1}{2\omega}(kAB\omega^3 - mAB\omega^2)\cos{2\omega t}[/tex]

    i am supposed to get something only in terms of A and B but it seems like it will since nothing really cancels out. did i make a mistake? help on this will be greatly appreciated.
     
  2. jcsd
  3. Aug 2, 2011 #2

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    I don't think the units on some of your terms work out, so you apparently made a math error somewhere. Recheck your calculations.
     
  4. Aug 2, 2011 #3
    for harmonic oscillator V(x)=1/2kx^2

    try maximizing E= V(x)+kinetic energy


    KE = 1/2(m)(dx/dt)^2

    its a classical solution but i think it still applies
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Calculating the energy of a harmonic oscillator
Loading...