Undergraduate Classical Mechanics Spring 2017

Physics 319

Classical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 23

Undergraduate Classical Mechanics Spring 2017

Hamiltonian Mechanics

• Built on Lagrangian Mechanics

• In Hamiltonian Mechanics

– Generalized coordinates and generalized momenta are

the fundamental variables

– Equations of motion are first order with time as the

independent variable

• In EE, very much like “state space” formalism

• You will see this trick similarly in relativistic

quantum mechanics when go from Klein-Gordon

type of equation to Dirac Equation

– Very general choices and transformations of the

coordinates and momenta are allowed

Undergraduate Classical Mechanics Spring 2017

General Procedure

1. Write down Lagrangian for the problem

2. Determine the generalized momenta

3. Determine the Hamiltonian function (energy for simple

problems) and write it in terms of the coordinates and

momenta

4. Solve equations of motion in Hamiltonian form

i

i

pq

L

1 1 1 1

1

, , , , , , , , , ,n

n n i i n n

i

q q p p p q q q p p

H L

Undergraduate Classical Mechanics Spring 2017

1-D Case

• General 1-D motion solvable

• Equations of motion

2

2 2 2

,2

/

2 2

A xx U x px x p

p A x x x p A xq

p p pU x U x

A x A x A x

L = H = L

L

H =

2

22

p dA Up

x A x dx x x

px

p A x

H L

H

Undergraduate Classical Mechanics Spring 2017

Simple Oscillator

• Lagrangian

• Generalized momentum

Also called the canonical momentum

• Hamiltonian

2 2

2 2

m kx xL =

2 2 2 2

2,2 2 2 2

p p k p kxq p x

m m m

H =

p mxx

L=

Undergraduate Classical Mechanics Spring 2017

Hamilton’s Equations of Motion

• Equations of motion in Hamiltonian form are

• General proof

• Argument works even when the Lagrangian/Hamiltonian

depends explicitly on time

dx dp

dt p dt x

px p kx

m

H H

, ,i i i i

i ii i i

i i ii

i ii i

i i i ii

q q q p t

q qq p q

p p pq

q qp p

q q q pq

H L

H L L

Undergraduate Classical Mechanics Spring 2017

Hamiltonian Conserved

• The time dependence of the Hamiltonian function is given

by

• When Langrangian/Hamiltonian does not explicitly depend

on time, the Hamiltonian (energy) is conserved

1

1

, ,i i

n

i i

i i i

n

i i i i

i

q p t

dq p

dt q tp

p q q pt t

H H

H H H H

H H

0d

dt

H

Undergraduate Classical Mechanics Spring 2017

Phase Space

• The set of variables describing the system in Hamiltonian

form is called phase space

Phase space variables for particle i

• Think of the motion occurring through phase space

,i iq p

Undergraduate Classical Mechanics Spring 2017

Central Force Again

• Hamiltonian equations of motion

• Ignorable coordinate gives conservation law. Reduction!

2 2 2

2 2 2

2

2 2 2

2

2

/

2

r

r

mT r r

mr r U r

p mr p mr

p p rU r

m

L

H =

2

3

2

0

rr

r

pp Ur p

p m r mr r

pp

p mr

H H

H H

Undergraduate Classical Mechanics Spring 2017

Example: Mass on a Cone

Undergraduate Classical Mechanics Spring 2017

Lagrangian and Hamiltonian

• Lagrangian

• Hamiltonian

2 2 2 2

2 2 2 2 2

2

12

mT r r z U mgz

r cz

mc z c z mgz

L

2 2 2 2

2 2

2 2 2 2 2

/

1 / 1

/ / 1

2

z z

z

p mc z p mc z

p m c z z p m c

p c z p cmgz

m

H

Undergraduate Classical Mechanics Spring 2017

Equations of Motion

• z direction

• θ direction

Conservation of angular momentum again. Centrifugal

barrier at z = 0

• Balanced condition

2 2 0

pp

p mc z

H H

22 2 3

2 3

pmg p m c gz

mc z

2

2 32

1

zz

z

ppz mg p

p z mc zm c

H H

Undergraduate Classical Mechanics Spring 2017

Ignorable Coordinates

• As in Lagrangian theory independence of the

Lagrangian/Hamiltonian on a coordinate guarantees a

conserved quantity

• In Hamiltonian theory, reduce the number of degrees of

freedom in the problem

• Simply evaluate conserved quantity using initial conditions

and substitute into Hamiltonian

• In most recent example

is a perfectly good 1-D potential for the z motion, once pθ

is evaluated using the initial conditions

2 2 2 2 2/ / 1

,2

z

z

p c z p cz p mgz

m

H