Application of Perturbation Theory in Classical Mechanics

1. Application of Perturbation Theory in Classical Mechanics - Shashidhar Guttula

2. Outline • Classical Mechanics • Perturbation Theory • Applications of the theory • Simulation of Mechanical systems • Conclusions • References

3. Classical Mechanics • Minimum Principles • Central Force Theorem • Rigid Body Motion • Oscillations • Theory of Relativity • Chaos

4. Perturbation Theory • Mathematical Method used to find an approximate solution to a problem which cannot be solved exactly • An expression for the desired solution in terms of a *power series

5. Method of Perturbation theory • Technique for obtaining approx solution based on smallness of perturbation Hamiltonian and on the assumed smallness of the changes in the solutions • If the change in the Hamiltonian is small, the overall effect of the perturbation on the motion can be large • Perturbation solution should be carefully analyzed so it is physically correct

6. Classical Perturbation theory • Time Dependent Perturbation theory • Time Independent Perturbation theory • Classical Perturbation Theory is more complicated than Quantum Perturbation theory • Many similarities between classical perturbation theory and quantum perturbation theory

7. Solve :Perturbation theory problems • A regular perturbation is an equation of the form : D (x; φ)=0 • Write the solution as a power series : • xsol=x0+x1+x2+x3+….. • Insert the power series into the equation and rearrange to a new power series in • D(xsol;”)=D(x0+x1+x2+x3+…..); =P0(x0;0)+P1(x0;x1)+P2(x0;x1;x2)+…. • Set each coefficient in the power series equal to zero and solve the resulting systems • P0(x0;0)=D(x0;0)=0 • P1(x0;x1)=0 • P2(x0;x1;x2)=0

8. Idea applies in many contexts • To Obtain • Approximate solutions to algebraic and transcendental equations • Approximate expressions to definite integrals • Ordinary and partial differential equations

9. Perturbation Theory Vs Numerical Techniques • Produce analytical approximations that reveal the essential dependence of the exact solution on the parameters in a more satisfactory way • Problems which cannot be easily solved numerically may yield to perturbation method • Perturbation analysis is often Complementary to Numerical methods

10. Applications in Classical Mechanics • Projectile Motion • Damped Harmonic Oscillator • Three Body Problem • Spring-mass system

11. Projectile Motion • In 2-D,without air resistance parameters • Initial velocity:V0 ; Angle of elevation :θ • Add the effect of air resistance to the motion of the projectile • Equations of motion change • The range under this assumption decreases. • *Force caused by air resistance is directly proportional to the projectile velocity

12. Force Drag k << g/V Effect of air resistance : projectile motion

13. Range Vs Retarding Force Constant ‘k’ from P.T

14. Damped Harmonic Oscillator • Taking • Putting

15. Harmonic Oscillator (contd.) • First Order Term • Second Order Term • General Solution through perturbation • Exact Solution

16. Three Body Problem • The varying perturbation of the Sun’s gravity on the Earth-Moon orbit as Earth revolves around the Sun • Secular Perturbation theory • Long-period oscillations in planetary orbits • It has the potential to explain many of the orbital properties of these systems • Application for planetary systems with three or four planets • It determines orbital spacing, eccentricities and inclinations in planetary systems

17. Spring-mass system with no damping

18. Input :Impulse Signal

19. Displacement Vs Time

20. Spring-mass system with damping factor

21. Input Impulse Signal

22. Displacement Vs Time

23. Conclusions • Use of Perturbation theory in mechanical systems • Math involved in it is complicated • Theory which is vast has its application • Quantum Mechanics • High Energy Particle Physics • Semiconductor Physics • Its like an art must be learned by doing

24. References • Classical Dynamics of particles and systems ,Marion &Thornton 4th Edition • Classical Mechanics, Goldstein, Poole & Safko, Third Edition • A First look at Perturbation theory ,James G.Simmonds & James E.Mann,Jr • Perturbation theory in Classical Mechanics, F M Fernandez,Eur.J.Phys.18 (1997) • Introduction to Perturbation Techniques ,Nayfeh. A.H