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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 |

language: | English |

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