Application of Perturbation Theory
in
Classical Mechanics
- Shashidhar Guttula
Outline
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Classical Mechanics
Perturbation Theory
Applications of the theory
Simulation of Mechanical systems
Conclusions
References
Classical Mechanics
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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
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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
R
(1  e  kT )
k
kV  g
T 
(1  e  kT )
gk
4kV
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 2 x dx
m 2  b  bo  k x  F
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