Undergraduate Classical Mechanics Spring 2017

Physics 319

Classical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 16

Undergraduate Classical Mechanics Spring 2017

• First determine the constant Laplace-Runge-Lenz vector

Elliptical Motion

2

2 2

1 2

2

2 2

1 2

1 2 1 2

1 2 1 2

2 2

0

cos sin sin

sin cos cos

cos sin cos sin cos

sin cos sin cos sin

ˆ ˆ ˆcos

x

y

x y x y

d l y l dr

dt r r Gm m dt

d l x l dr

dt r r Gm m dt

l d lA r r r

Gm m dt Gm m

l d lA r r r

Gm m dt Gm m

A x A y A A

2 2

0ˆsinx yx A A y

0

2 2

x yA AyA

xA

Undergraduate Classical Mechanics Spring 2017

2

2

1 2

2 2

0

2

1 2 m

2

min 0

02

1 2

ˆ ˆ

cos sin

cos

/

1 1 / cos21 1 cos

x y

x y

L lz r z

lr A r A r

Gm m

r A A

l Gm m rr

E EEl

Gm m

2 2 22 2 2 2 2

2 2 2

1 2 1 2 1 2 1 2

21 2 1x y

l l l ElA A r r r

Gm m Gm m Gm m Gm m

• Determine the vector magnitude

• Equation for radius r(θ)

Orbit as function of θ

Undergraduate Classical Mechanics Spring 2017

Parametric Equation for Ellipse

• Sum of the lengths to the two foci constant 2L

• Eccentricity is ε = f/L =

r b

ff

a

2 2 2 2

2 2 2

max min

cos 2 2 cos

sin 2 sin

4 cos 4 4 4

/

cos 1 / cos

r f L r

r L r

r r f f L Lr r

L f b Lr

L f f L

r L f r L f L a

2 21 /b a

aphelion perihelion

mr

Undergraduate Classical Mechanics Spring 2017

More on Ellipse

• In Homework show using Cartesian coordinates centered

on dot get the ellipse equation (Kepler’s First Law)

• Ellipse orientation in plane handled by θ0

2 2

2 21

x f yf a

a b

0

2

0

/

1 cos

b ar

2 /

1 cos

b ar

Undergraduate Classical Mechanics Spring 2017

Method Two

• Most common analysis, integrate the energy function

• Check: 2D harmonic oscillator

22 2 2

2 2 2 2

02 2 2 2

2 / 2 / /

2 / 2 / 1/

2 / 2 / 1/

dr dr dt rE U r l r

d dt d l

drd

r E l U r l r

dr

r E l U r l r

2

02 2 2 2 2

2 2 3

/ 2

2 / / 1/

1/ / 2 /

U r kr

dr

r E l kr l r

x r E l dx dr r

Undergraduate Classical Mechanics Spring 2017

Linear Restoring Force

• Equation for ellipse centered at origin

semi-major axis semi-minor axis

02

2 2 2

2 21

02 2 4 2

2 2 2 4 2

02

2 2 2 2 2

0 0

2 2 4 2 2 2 2 2

0 0

2

/ /

1/ /cos 2

/ /

1/ / / cos 2

/ cos sin

/ / cos sin 1

dx

E l k l x

r E l

E l k l

E l E l k lr

E l r r

E l k l r r

1/2 1/2

2 2 2 4 2 2 2 2 4 21/ / / / 1/ / / /E l E l k l E l E l k l

Undergraduate Classical Mechanics Spring 2017

Period

• Use area result. Total area of orbit ellipse

• Time derivative of area swept out by radius vector

• Period

• Works!

1/2 1/2

2 2 2 4 2 2 2 2 4 2/ / / / / /

ab

E l E l k l E l E l k l

k

2

dA l

dt

2

2 2

/ 2 //

abT

l kk l l

Undergraduate Classical Mechanics Spring 2017

Kepler Problem

• Case of gravitational attraction

1 2

02 2 2 2

1 2

2 2

1 2

02

2 2 2

1 2

2

1 20

22 2

1 2

2

1 2 m

2

min 0

02

1 2

/

2 / 2 / 1/

1/ / /

2 / /

1/ /cos

2 / /

/

1 1 / cos21 1 cos

U r Gm m r

dr

r E l Gm m rl r

x r Gm m l dx dr r

dx

E l Gm m l x

r Gm m l

E l Gm m l

l Gm m rr

E EEl

Gm m

Undergraduate Classical Mechanics Spring 2017

Method Three (Taylor)

• Write a differential equation for u = 1/r

• For Newton gravity

• Not so “attractive” because need to write a differential

equation for 1/r2 for linear restoring force

2

3 2

2

2 2

2 3

2

2 2 2

,

1 1

1

1/

U l dr dr l drr r r t t

r r dt d r d

du dru r

r d r d l

d u U lr

d l l r r

d u Uu r u

d l u r

2

1 2

2 2

Gm md uu

d l

Undergraduate Classical Mechanics Spring 2017

Relations Among Parameters

• For gravitational attraction and bound orbit

m

min 0

min min

2

1 2m 1 2

2 2

22

1 2

m

2

1 1 / cos

1 / / 0

/

1 1 2

1

1

rr

E E

E E E E

l Gm mr Gm ma

E

la Gm m

rb

Undergraduate Classical Mechanics Spring 2017

Kepler’s Third Law

• Revolution period T is

• Only power laws for the potential with closed (elliptical)

orbits are linear restoring force (kr2) and

gravitation/Coulomb force (k/r)

2 4 2 2

2

2

22

1 2

2 3 2 32

1 2 1 2

4 12

1

4 4

aabT T

l l

la Gm m

a aT

Gm m G m m

Undergraduate Classical Mechanics Spring 2017

Unbound Motion

• E = 0 means the motion is a parabola with focus at sun.

Minimum radius is rm/2

• For E > 0, motion is a hyperbola, ε > 1. Orbit asymptotes

are

• Minimum radius

0

1

0

0

1cos

cos 1/

/ 2

asymp

asymp

asymp

mmin

1

rr

Undergraduate Classical Mechanics Spring 2017

Impact Parameter

• Distance of closest approach for particle without

interacting is called the impact parameter bim

• Equation for hyperbola

• Equation for asymptotes

• Distance of closest approach and

eccentricity

asymp

m

2 1

r

m

1

r

imb

2

2 222 m

2 2 2

m m

111

1

ry x

r r

2 2

m m

21 2

2 1 1

1

imim

im

Ebr rb

b Gm m

2 m

21

1

ry x