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Class #12 of 30

Finish up problem 4 of examStatus of courseLagrange’s equations Worked examples

Atwood’s machine

:

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Falling raindrops redux II

1) Newton

2) On z-axis

3) Rewrite in terms of v

4) Rearrange terms

5) Separate variables

2

2

2

2

2

2

2

ˆ

Assume vertical motion

(1 )

(1 )

zz

z z

z

z

mr mgz cr

mz mg cz

dv cg v

dt m

dv vmgDefine v g

c dt v

dvg dt

v

v

dragF

gm

z

x

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Falling raindrops redux III

2

2(1 )

z

z

dvg dt

v

v

( )

20 0

2(1 )

v t tz

z

t

dvg dt gt

vv

( ) /

20arctanh( ( ) / )

(1 )

( ) tanh( / )

zz

v t v

vu dv v du

v

v duv v t v gt

u

v t v gt v

2arctanh( )

(1 )

dxx

x

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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

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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

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Joseph LaGrangeGiuseppe Lodovico Lagrangia

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.

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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]

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Lagrange’s Equation

Works best for conservative systemsEliminates the need to write down forces

of constraintAutomates the generation of differential

equations (physics for mathematicians)Is much more impressive to parents,

employers, and members of the opposite sex

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0i i

dF ma

q dt q

L L

T U L

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 equationsReduce 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) Calculate6) Plug into7) Solve ODE’s 8) Substitute back original variables

Lagrange’s Kitchen

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0i i

d

q dt q

L L

iq

T U L ,i iq q

Mechanics “Cookbook” for Lagrangian Formalism

dand and

q q dt q

L L L

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Degrees of Freedom for Multiparticle Systems

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5-N for multiple rigid bodies3-N for multiple particles

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Atwood’s MachineReverend 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.

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11

m1 m2

Atwood’s MachineLagrangian recipe

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2 21 1 2 2 1 1 2 2

1 11) ;

2 2T m y m y U m gy m gy

1 1 1 1 2 2 2 2

1 2 2 1 2 1

2) 2 10

, , , .; , , , .

particles degrees of freedom

x z const x z const

y y k y k y y y

21 2 1 2

14) ( ) ( )

2m m q g m m q L

1 1 2 23) , ; ;q y q y y k q y q

6) 0d

q dt q

L L

1 2

1 2 1 2

5) ( )

( ) ; ( )

g m mq

dm m q m m q

q dt q

L

L L

1 2 1 2( ) ( ) 0g m m m m q

T U L

1y2y

12

m1 m2

Atwood’s MachineLagrangian recipe

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21 2 1 2

14) ( ) ( )

2m m q g m m q L

6) 0d

q dt q

L L1 2 1 2( ) ( ) 0g m m m m q

*1 2

1 2

* 2 1 * 2 1 10 0 0 0

( )

( )

1 1

2 2

g m mq a

m m

q a t q t q y a t y t y

T U L

1y2y

1 1 2 23) , ; ;q y q y y k q y q

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Atwood’s MachineSimulation

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The simplest Lagrangian problem

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g

m

0

2 2 2

ˆ;

1 1 1

2 2 2

Mass m Velocity v x

U mgz

T mx my mz

A ball is thrown at v0 from a tower of

height s.

Calculate the ball’s subsequent motion

v0

1) Write down T and U in any convenient coordinate system.

2) Write down constraint equations3) Define the generalized coordinates 4) Rewrite in terms of 5) Calculate6) Plug into7) Solve ODE’s 8) Substitute back original variables 0

i i

d

q dt q

L L

iqT U L ,i iq q

dand and

q q dt q

L L L

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Class #12 Windup

Office hours today 4-6 Wed 4-5:30

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0i i

dF ma

q dt q

L L

T U L

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Atwood’s Machine with massive pulleyLagrangian recipe

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m1 m2

R

32 2 2

1 2 1 2

1 1 11) ;

2 2 2T m x m y I U m gx m gy

1y2y

1 1 1 1 2 2 2 2

13 3 3 3 3

1 2 2 1 2 1

2) 3 15

, , , .; , , , .

, , , .;

objects degrees of freedom

x z const x z const

yx y z const

Ry y k y k y y y

1 1 2 23) , ; ;q y q y y k q y q