simple harmonic oscillator - classical mechanics
DESCRIPTION
A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.TRANSCRIPT
Classical MechanicsA Presentation
OnLinear Harmonic Oscillator
Khulna UniversityMathematics Discipline
A relation of Lagrange’s equation of motion with simple harmonic motion
Lagrange’s equation of motion for one dimensional motion (at x direction ) is:
Moving through x axes
The kinetic energy of this system is :
The potential energy of this system is:
Here c is constant of integration and k is spring constant.
We know:
• A horizontal plane passing through the position of equilibrium:
If we choose the horizontal plane passing through the position of equilibrium as the reference level, then V=0 at x=0 so that c=0
So the Lagrangian is:
So that
And
Then we get from the Lagrange’s eqn :
Or,
It is an equation of simple harmonic motion and can be put in the form
Now in
We saw that the equation of simple harmonic motion can obtained from Lagrange’s motion of equation.
Reference:
Classical Mechanics : by Gupta Kumar Sharma 14th Edition : Chapter 1
Internet : (Wikipedia, Mathforum)
Presented by:
Debashis BaidyaStudent ID : 11124911 batch
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