Simple harmonic oscillator - Classical Mechanics

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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