Undergraduate Classical Mechanics Spring 2017

Physics 319

Classical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 14

Undergraduate Classical Mechanics Spring 2017

Lagrangian Method

1. Choose coordinates to incorporate constraint

automatically.

2. Write down the Euler-Lagrange Equation.

3. Solve the Euler-Lagrange Equation.

• Double Pendulum example

• Analyzed in detail in Chapter 11

(0,0)

1 1 1 1 1 1

2 2 1 1 2 2 2 2

2 2 2 2 2 21 2 21 2 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 2 2

1 1 1 2 1 1 2 2 2

, cos , sin

, , cos , sin

, , , sin sin cos cos2 2 2

cos cos cos

x y l l

x y x y l l

m m ml l m l l l

m gl m gl m gl

L

l1

l2

1

2

Undergraduate Classical Mechanics Spring 2017

Ignorable or Cyclic Coordinates

• Lagrangian for particle acted on by gravity

• Does not depend on x or y. Euler-Lagrange imply

• When the Lagrangian is independent of a generalized

coordinate that coordinate is said to by ignorable or cyclic.

Automatically, generalized momentum is conserved.

Usually this degree of freedom can be integrated completely

using the conservation of generalized momentum

2 2 2

2

mx y z mgz L

0 0x y

d dmx p C my p C

dt dt

0 0ii

i i

dpdp C

q dt q dt

L L

Undergraduate Classical Mechanics Spring 2017

Conservation Laws/Nöther’s Theorem

• Suppose Langrangian is translationally invariant

This invariance in Lagrangian implies momentum conserved

• Example: N interacting bodies

, ,

0 0i i ii i i

q q q q

dC

q dt q q

L L

L L L L

12

int

1 1 12

, ,

N N ii

i i j

i i j

j j j j

mv U r r

v r v r

L

L L

Undergraduate Classical Mechanics Spring 2017

Momentum Conservation

• Total momentum conserved

• When Lagrangian invariant to (infinitesimal) rotation

1 1

int,

1 1 1 1

0N N i i

i x i i i

i i j j

Udm v x x

dt x

, ,

, , ,

, , ,

,

ˆ ˆ, ,

0 0

i i i i

i i jk j i k i i i jk j i k i i i

i jk i k i jk i k i jk i k

i i ij i i i i i i

r r r r

r r x r r x r r

dr r r

r r dt r

L

L L

L L L L

Undergraduate Classical Mechanics Spring 2017

Angular Momentum Conservation

• Total angular momentum conserved

• When Lagrangian independent of time

, , 0i i i i jk i k

i

j i i ij

i

dm r r

dt

L r m r C

1 1 1 1

1 1

1 1 1

, , , , , , , , , , , ,

0

n n n n

n n

i i

i ii i

n n n

i i

i i ii i i

q q q q t q q q q t

d dq q

t dt q dt q

d d d dq q

dt q dt dt q dt q

L L

L L L L L

L L L LL

Undergraduate Classical Mechanics Spring 2017

Hamiltonian Function

• The negative of the preceding combination is called the

Hamiltonian function

• The Hamiltonian is conserved

• Taylor shows for time-independent generalized coordinate

transformations, leading to quadratic dependence of the

Lagrangian on the

• Energy conserved!

1

n

i

i i

qq

L

H L

2T T U T U H

iq

Undergraduate Classical Mechanics Spring 2017

Lagrangian for Electromagnetic Force

• Electromagnetic Lorentz force follows from the

Lagrangian

• Canonical momentum

2 2 2

2x y z

mx y z q q xA yA zA L

mr qA Px

L

yx x x x x z

d A Amr qA mr q r

dt t x xx

AA A A A A Amx q qx qy qz q qx qy qz

t x y z x x x x

L L

P

Undergraduate Classical Mechanics Spring 2017

y y y y yx z

yxz z z z z

y x x zx

A A A A AA Amy q qx qy qz q qx qy qz

t x y z y y y y

AAA A A A Amz q qx qy qz q qx qy qz

t x y z z z z z

A A A Amx qE qy qz

x y z x

my q

y y xzy

yx z zz

A A AAE qz qx

y z x y

AA A Amz qE qx qy

z x y z

ma q E v B

Undergraduate Classical Mechanics Spring 2017

Lagrange Multipliers

• Sometimes constraints are not easily accounted for only by

coordinate choice. If the constraint can be put in the form

of a time-integral along the path, the problem can be

solved using Lagrange multipliers

• Find stationary condition using Euler-Lagrange for

• Works because the constrained variation of the added λf is

zero (it integrates to a constant) and does not change the

fact that the constrained solution is still has stationary

Lagrangian

1 1

1 1

, , , , , ,

, , , , , ,

n n

n n

S q q q q t dt

C f q q q q t dt

L

S f dt L

Undergraduate Classical Mechanics Spring 2017

Snell’s Law 3!

• In terms of two angles

1 1 2 2

1 2

1 1 2 2 2

1 1 2 21 1 2 2

1 2

2

2

1 1 2 2

cos cos

tan tan

tan tan 0cos cos

sinsec 0 sin

cos

sin sin

tot

i

i i ii i i i

i

n h n hL

h h x

n h n hh h

n hh n

n n