Simple harmonic oscillator - Classical Mechanics
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The document presents a mathematical overview of the linear harmonic oscillator, relating Lagrange's equation of motion to simple harmonic motion. It discusses the kinetic and potential energy of the system, leading to the derivation of the equation of simple harmonic motion. The presentation is based on 'Classical Mechanics' by Gupta Kumar Sharma and includes contributions from various sources.
Simple harmonic oscillator - Classical Mechanics
- 1. Classical Mechanics A Presentation On Linear Harmonic Oscillator Khulna University Mathematics Discipline
- 2. 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: 𝑑 𝑑𝑡 𝜕𝐿 𝜕𝑥 − 𝜕𝐿 𝜕𝑥 = 0
- 3. Moving through x axes
- 4. The kinetic energy of this system is : 𝑇 = 1 2 𝑚 𝑥2 The potential energy of this system is: 𝑉 = − 𝑓. 𝑑𝑥 = − −𝑘𝑥𝑑𝑥 = 1 2 𝑘𝑥2 + 𝑐 Here c is constant of integration and k is spring constant. We know: 𝑓 = −𝑘𝑥
- 5. • 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
- 6. So the Lagrangian is: 𝐿 = 1 2 𝑚 𝑥2 − 1 2 𝑘𝑥2 So that 𝛿𝐿 𝛿 𝑥 = 𝑚 𝑥 And 𝛿𝐿 𝛿𝑥 = −𝑘𝑥
- 7. Then we get from the Lagrange’s eqn : 𝑑 𝑑𝑡 𝑚 𝑥 − (−𝑘𝑥) = 0 Or, 𝑚 𝑥 + 𝑘𝑥 = 0 It is an equation of simple harmonic motion and can be put in the form 𝑥 + 𝜔2 𝑥 = 0
- 8. Now in 𝑥 + 𝜔2 𝑥 = 0 𝜔 𝑖𝑠 𝑡ℎ𝑒 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑜𝑓 𝑜𝑠𝑐𝑖𝑙𝑙𝑎𝑡𝑖𝑜𝑛 We saw that the equation of simple harmonic motion can obtained from Lagrange’s motion of equation.
- 9. Reference: Classical Mechanics : by Gupta Kumar Sharma 14th Edition : Chapter 1 Internet : (Wikipedia, Mathforum)
- 10. Presented by: Debashis Baidya Student ID : 111249 11 batch
- 11. THANK YOU