Application of Perturbation Theory in Classical Mechanics Shashidhar

Application of Perturbation Theory in Classical Mechanics - Shashidhar Guttula

Outline • • • Classical Mechanics Perturbation Theory Applications of 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=x 0+x 1+x 2+x 3+…. . – Insert the power series into the equation and rearrange to a new power series in • D(xsol; ”)=D(x 0+x 1+x 2+x 3+…. . ); =P 0(x 0; 0)+P 1(x 0; x 1)+P 2(x 0; x 1; x 2)+…. – Set each coefficient in the power series equal to zero and solve the resulting systems • P 0(x 0; 0)=D(x 0; 0)=0 • P 1(x 0; x 1)=0 • P 2(x 0; x 1; x 2)=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: V 0 ; 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

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

Input : Impulse Signal

Displacement Vs Time

Spring-mass system with damping factor

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 4 th 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