- #1
bluesquare
- 4
- 0
Homework Statement
Okay, I am trying to solve this Anharmonic Oscillator equation. Now I am given with the potential
[tex]
U=(1/2)x^2-(1/4)x^4
[/tex]
and Kinetic energy
[tex]
T=(1/2)x' ^2
[/tex]
So the Lagrangian becomes
[itex]\mathcal L[/itex][tex]=T-U[/tex]
Now I have taken all the [tex]k[/tex]'s and [tex]m[/tex] to be 1
Homework Equations
After solving the Lagrangian Equation I got
[tex]
x''(t)=-x(t)+x(t)^3
[/tex]
The Attempt at a Solution
And when I used the solution [tex]x(t)=tanh(t/\sqrt{2}) [/tex] it seems to be satisfying. But my problem is to find a general solution in where I can express my total energy as initial value condition by mentioning my energy and there by controlling how the system behaves e.g how it's position and velocity depends on total energy sort of like SHO problem where [tex]x(t)=\sqrt{2E}sin (t)[/tex] and [tex]v(t)=\sqrt{2E}cos (t)[/tex]
I want my Anharmonic Oscillator position and velocity to be represented like this where [tex]E[/tex] and [tex]t[/tex] clearly providing the initial conditions.
Thank you for the time.