Class #15

1. Class #15 • Course status • Finish up Vibrations • Lagrange’s equations • Worked examples • Atwood’s machine • Test #2 11/2 :

2. Physics Concepts • Classical Mechanics • Study of how things move • Newton’s laws • Conservation laws • Solutions in different reference frames (including rotating and accelerated reference frames) • Lagrangian formulation • Central force problems – orbital mechanics • Rigid body-motion • Oscillations • Chaos :

3. Mathematical Methods • Vector Calculus • Differential equations of vector quantities • Partial differential equations • More tricks w/ cross product and dot product • Stokes Theorem • “Div, grad, curl and all that” • Matrices • Coordinate change / rotations • Diagonalization / eigenvalues / principal axes • Lagrangian formulation • Calculus of variations • “Functionals” • General Mathematical competence :

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5. Joseph LaGrangeGiuseppe Lodovico Lagrangia “The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.”Preface to Mécanique Analytique. Joseph Lagrange [1736-1813]  (Variational Calculus, Lagrangian Mechanics, Theory of Diff. Eq’s.) Greatness recognized by Euler and D’Alembert 1788 – Wrote “Analytical Mechanics”. You’re taking his course. Rescued from the guillotine by Lavoisier – who was himself killed. Lagrange Said:“It took the mob only a moment to remove his head; a century will not suffice to reproduce it.” “If I had not inherited a fortune I should probably not have cast my lot with mathematics.” “I do not know.” [summarizing his life's work] :45

6. Generalized Force and Momentum Traditional Generalized Force Torque Linear Momentum Angular Momentum Newton’s Law

7. Lagrange’s Equation Works for conservative systems Eliminates need to show forces of constraint Requires that forces of constraint do no work Requires the clever choice of q consistent w/ forces of constraint. Requires unique mapping between Lagrangian must be written down in inertial frame Automates the generation of differential equations (physics for mathematicians)

8. Lagrange’s Kitchen Mechanics “Cookbook” for Lagrangian Formalism • Write down T and U in anyconvenient coordinate system. 2) Write down constraint equations Reduce 3N or 5N degrees of freedom to smaller number. 3) Define the generalized coordinates Consistent with the physical constraints 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables

9. Degrees of Freedom for Multiparticle Systems • 5-N for multiple rigid bodies • 3-N for multiple particles

10. Atwood’s MachineLagrangian recipe m1 m2

11. Atwood’s MachineLagrangian recipe m1 m2

12. T7-17 Atwood’s Machine with massive pulleyLagrangian recipe R I m1 m2 1) Write down T and U in anyconvenient coordinate system. 2) Write down constraint equations 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables

13. Atwood’s Machine with massive pulleyLagrangian recipe R I m1 m2

14. The simplest Lagrangian problem 1) Write down T and U in anyconvenient coordinate system. 2) Write down constraint equations 3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables v0 m A ball is thrown at v0 from a tower of height s. Calculate the ball’s subsequent motion g