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