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

)3

41(

)1(

)1(

gkVRR

egkgkVT

ekUR

o

kT

kT

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

Fkxdtxdm 2

2

Input :Impulse Signal

Displacement Vs Time

Spring-mass system with damping factor

Fxkbdtdxb

dtxdm o 2

2

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