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

2 : 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 Physics Concepts

3 : 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 Mathematical Methods

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5 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]

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 1) Write down T and U in any convenient 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 Lagrange’s Kitchen Mechanics “Cookbook” for Lagrangian Formalism

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

10 m1m1 m2m2 Atwood’s Machine Lagrangian recipe

11 m1m1 m2m2 Atwood’s Machine Lagrangian recipe

12 T7-17 Atwood’s Machine with massive pulley Lagrangian recipe m1m1 m2m2  R 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

13 Atwood’s Machine with massive pulley Lagrangian recipe m1m1 m2m2  R

14 The simplest Lagrangian problem 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