Lecture 18 Outline: Symmetric Top and Oscillations - Prof. M. W. Bromley, Study notes of Physics

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• Lecture 18 Outline - transit 5 to 6

Symmetric Top example (Section 5.7)

Oscillations, theory of (Section 6.1/6.2)

Euler Equations of Motion contd

last time... gave us Euler’s equations of motion:

I

1

˙ω 1 − ω 2 ω 3 ( I 2 − I 3

N

1

I

2

˙ω 2 − ω 3 ω 1 ( I 3 − I 1

N

2

I

3

˙ω 3 − ω 1 ω 2 ( I 1 − I 2

N

3

Continuing the dumbbell example, if

ω

then

given

ω

and remembering

~ω

= [

, ω

sin

(^) θ, ω

cos

(^) θ

]

Then only one Torque component is non-zero N x = − ω y ω z

I

2

−

I

3 ) =

− ( m 1 + m 2 ) b 2 ω 2

sin

θ

cos

(^) θ

ie spins the dumbbell around in the

x

-direction

Symmetrical Top - Eulerian Way

For gyroscope and spinning top

ψ

θ

φ

now

I 1 = I 2 6

I

3

(since symmetry along

z )

from Euler’s eqns, torque with

N

1 , N

2

won’t change

ω

3

N 3 = I 3 ˙

ω 3 − ω 1 ω 2 ( I 1 − I 2

I

3

˙ω 3

initial torque

N

3

=

N

2

= 0

only

N

1

6 = 0

and initial rotation

ω

1

=

ω

2

= 0

only

ω

3

6 = 0 =

constant

But time evolution has

ω

1

and

ω

2

changing

ie complicated motion in terms of

ψ,

φ,

θ˙

Symmetrical Top - Lagrangian Way

A (somewhat) easier solution from the Lagrangian...

T = T 1 + T 2 + T 3 =

2 1 (^) I 1 ( ω

(^12)

ω

(^22) (^) ) +

2 1 (^) I 3 ω

(^32)

Now, Goldstein 4.87 has exactly these 3

ω

’s in terms of

~ω

ω

x ′

ω

y ′

ω

z ′  

φ

(^) sin

(^) θ

sin

(^) ψ

θ˙

cos

ψ

φ

(^) sin

(^) θ

cos

(^) ψ

θ˙

sin

(^) ψ

φ

cos

(^) θ

ψ

and in terms of Euler angle’s

V

mgl

cos

(^) θ

and

T

I

1

θ˙ 2

φ

2

sin

2 θ ) + I 3

ψ

φ

(^) cos

(^) θ

)

2

giving a Lagrangian with two cylic coords (

φ, ψ

ie the

p

φ

, p

ψ

components are constant in time

Applications

Given that the

ω

3

is constant in time

this means that the fourth constant of motion really is

E

′

E

I

3 ω

(^32)

I

1

θ 2

I

1

b

−

a

cos

(^) θ

) 2

sin

2

θ

mgl

(^) cos

(^) θ

I

1

˙

θ 2

V

′ ( θ )

this can be solved for

θ ( t )

, but it’s complicated

darkness (ie. elliptical integrals).we won’t solve it here since it just leads down into

Goldstein 5.7There are limited regimes of soln see remainder

mounted such that centre of mass is fixed...Gyroscopes pg 222,223... are rapidly rotating devices

Oscillations formalism - intro

interested in motion about equilibrium

system is in equilibrium when generalised forces

Q

i

=

∂q ∂V

i )

0

= 0

ie extremum at equilibrium

q 01

, q

02

,... , q

on

As per usual equilibrium classified stable/unstable

equilibrium:in terms of the deviations of system energy from

eg. 2-D isotropic SHO

Oscillations formalism - kinetic

In the distant past

~v i

=

d~ r i

dt

j

∂~ r i

∂q

j

˙

q j

∂~ r i

∂t

(see Goldstein equations 1.71 and 1.72 on page 25)

our generalised coords don’t explicitly depend on time:

T

i

m

i v i 2

=

i

m

i (

j

r i

∂q

j

˙q j )

2

=

i,j

M

ij

q i ˙

q j

The coeffs

M

ij

are functions of

q k ,

M

ij (^) ( q 1 , q

2 , .., q

n ) =

m

ij (^) ( q 01

, q

02

, .., q

on

∂m

ij

∂q

k ) 0 η k +

only include constants since

T

has quadratic terms in

˙q i

so we have

M

ij

m

ij (^) ( q 01

, q

02

, .., q

on

T

ij

and

T

m

ij

η i ˙

η j

=

T

ij

η i ˙

η j