Application of Perturbation Theory in Classical Mechanics

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					Application of Perturbation Theory
                 in
       Classical Mechanics


                 - Shashidhar Guttula
Outline
•   Classical Mechanics
•   Perturbation Theory
•   Applications of the theory
•   Simulation of Mechanical systems
•   Conclusions
•   References
Classical Mechanics
•   Minimum Principles
•   Central Force Theorem
•   Rigid Body Motion
•   Oscillations
•   Theory of Relativity
•   Chaos
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
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
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
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
Idea applies in many contexts
• To Obtain
  – Approximate solutions to algebraic and
    transcendental equations
  – Approximate expressions to definite
    integrals
  – Ordinary and partial differential equations
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
Applications in Classical Mechanics

•   Projectile Motion
•   Damped Harmonic Oscillator
•   Three Body Problem
•   Spring-mass system
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
Force Drag k << g/V
 Effect of air resistance : projectile motion

             U          kT
         R     (1  e      )
             k
             kV  g             kT
         T           (1  e        )
               gk
                     4k V
         R  Ro (1          )
                      3g
Range Vs Retarding Force Constant ‘k’ from P.T
Damped Harmonic Oscillator



• Taking




• Putting
Harmonic Oscillator (contd.)

• First Order Term

• Second Order Term

• General Solution through perturbation



• Exact Solution
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
Spring-mass system with no damping




                          d 2x
                         m 2  kx  F
                          dt
Input :Impulse Signal
Displacement Vs Time
Spring-mass system with damping factor




                             d 2x
                            m 2  b  bo  k x  F
                                   dx
                             dt    dt
Input Impulse Signal
Displacement Vs Time
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
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

				
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posted:11/17/2011
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