Class #12 of 30

1. Class #12 of 30 • Finish up problem 4 of exam • Status of course • Lagrange’s equations • Worked examples • Atwood’s machine :

2. Falling raindrops redux II • Newton 2) On z-axis 3) Rewrite in terms of v 4) Rearrange terms 5) Separate variables z x :45

3. Falling raindrops redux III :50

4. 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 (and Hamiltonian form.) • Central force problems – orbital mechanics • Rigid body-motion • Oscillations • Chaos :04

5. 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” • Lagrange multipliers for constraints • General Mathematical competence :06

6. 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 :08

7. Lagrange’s Equation Works best for conservative systems Eliminates the need to write down forces of constraint Automates the generation of differential equations (physics for mathematicians) Is much more impressive to parents, employers, and members of the opposite sex :12

8. Lagrange’s Kitchen Mechanics “Cookbook” for Lagrangian Formalism • Write down T and U in anyconvenient coordinate system. It is better to pick “natural coords”, but isn’t necessary. 2) Write down constraint equations Reduce 3N or 5N degrees of freedom to smaller number. 3) Define the generalized coordinates One for each degree of freedom 4) Rewrite in terms of 5) Calculate 6) Plug into 7) Solve ODE’s 8) Substitute back original variables :17

9. Degrees of Freedom for Multiparticle Systems • 5-N for multiple rigid bodies • 3-N for multiple particles :20

10. Atwood’s MachineReverend George Atwood – Trinity College, Cambridge / 1784 Two masses are hung from a frictionless, massless pulley and released. • Describe their acceleration and motion. • Imagine the pulley is a disk of radius R and moment of inertia I. Solve again. :45 :25

11. Atwood’s MachineLagrangian recipe m1 m2 :40

12. Atwood’s MachineLagrangian recipe m1 m2 :45

13. Atwood’s MachineSimulation :45 :50

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 :65

15. Class #12 Windup • Office hours today 4-6 • Wed 4-5:30 :72

16. Atwood’s Machine with massive pulleyLagrangian recipe R I m1 m2 :70