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 1) 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 LaGrange Giuseppe Lodovico Lagrangia Joseph Lagrange [ ]  (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. :45 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. 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] :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 1) Write down T and U in any convenient 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 Lagrange’s Kitchen :17 Mechanics “Cookbook” for Lagrangian Formalism

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

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

11 m1m1 m2m2 Atwood’s Machine Lagrangian recipe :40

12 m1m1 m2m2 Atwood’s Machine Lagrangian recipe :45

13 Atwood’s Machine Simulation :45 :50

14 The simplest Lagrangian problem :65 g m A ball is thrown at v 0 from a tower of height s. Calculate the ball’s subsequent motion v0v0 1) Write down T and U in any convenient 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

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

16 Atwood’s Machine with massive pulley Lagrangian recipe :70 m1m1 m2m2  R