1

Class #15

Course statusFinish up VibrationsLagrange’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

4 :

5

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.

: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

i i

d

dF P

t

x

UF

x

i i

i i

p mv

L I

Generalized Force and MomentumTraditional Generalized

i i

d

dt qq

LL

Force

Torque

Linear Momentu

m

Angular Momentu

m

Newton’s Law

T U L

ii

L

2

2

1

2

1

2

i ii

i ii

p mxx

L I

ii

pq

L

ii

Fx

L

ii

Fq

L

F U

U

7

Lagrange’s Equation

Works for conservative systemsEliminates need to show forces of constraint

Requires that forces of constraint do no workRequires the clever choice of q consistent w/ forces of constraint. Requires unique mapping between

Lagrangian must be written down in inertial frameAutomates the generation of differential equations

(physics for mathematicians)

0 0i i

dF ma

q dt q

L L

1 2 1 2( , ... ) ( , ... )i n i nr r q q q and q q r r r

8

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

2) Write down constraint equationsReduce 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) Calculate6) Plug into7) Solve ODE’s 8) Substitute back original variables

Lagrange’s Kitchen

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

9

Degrees of Freedom for Multiparticle Systems

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

10

m1 m2

Atwood’s MachineLagrangian recipe

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

11

m1 m2

Atwood’s MachineLagrangian recipe

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

0 0

*1 2

1 2

* 20 0

* 21 1 1

( )

( )

17)

21

8)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

12

T7-17 Atwood’s Machine with massive pulley

Lagrangian recipe

1 1 1 1 2 2 2 2

3 3 3 3 3

1 2

2) 3

, , , .; , , , .

, , , .; ?

, ?

objects degrees of freedom

x z const x z const

x y z const

y y constraint

3) ??q m1 m2

R

3

1y2y

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

1) ;T U

13

Atwood’s Machine with massive pulleyLagrangian recipe

m1 m2

R

3

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

2 2 21 1 2 2 3 3 1 1 2 2

1 1 1

2 2 21) ;T m y m y I U m gy m gy

14

The simplest Lagrangian problem

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