physics 451/551 theoretical mechanics...theoretical mechanics fall 2018 action-angle variables •...

Theoretical Mechanics Fall 2018

Physics 451/551

Theoretical Mechanics

G. A. Krafft

Old Dominion University

Jefferson Lab

Lecture 11


Action-Angle Variables

• Suppose a mechanical system has a periodic motion

– Libration (return in phase space)

– Rotation (oscillating momentum)

• In some situations knowledge of the full motion is not so

interesting as knowing the frequencies of motion in the

system

• Frequencies determined by the following procedure

1. Define the action

2. Determine the Hamiltonian as a function of action

3. Frequencies are the derivatives

no summationi

i i

H C

J p dq

, ,i nH J J


Real Pendulum Phase Space

• Hamiltonian

• Action is the phase space area

2

2cos

2

PH mgL

mL

p

H


Relationship to period

• Action is

• Derivative with respect to energy is

The RHS is simply the oscillation period

0

0

1

02 2 cos cosJ mL mgL dmgL

0 0

0 0

2 22 2

22 cos / 2

dJ mL mL dd T

dE mgL m L


More General Argument

• Classical Action for Motion

• Oscillation action

Depends only on α, not q

• Invert to get

, /S W q t p W q

1

,

H J f J

S W q H J H J t

0 02 , ,J W W f


Angle Variable

• Define “angle” variable

• Constant of motion

Angle variable increases linearly with time. Again

gives the frequency

• Fetter and Walecka have generalization for many

“separable” degrees of freedom

Ww

J

Sw t

J J

/H J


Symplectic Matrices

• Assume the even-dimensional manifold (and vector space)

R2n. A matrix acting on vectors in R2n is called symplectic

if it preserves the canonical symplectic structure

• Such matrices form a matrix Lie group (like rotations!)

2 2

1

, , ,

, ,

ni

i

i

S S dp dq S S

S S

1 2 1 2 2 2

1 1 1 1

, , ,

, , ,

S S S S S S

S S SS SS


Symplectic Condition

• Note (co-ordinate convention (q1,…, qn,p1,…,pn))

• Symplectic means

• Determinate is always +1

• Another definition of canonical transformation: a

symplectic matrix

0,

0

t t

t

IS S S S

I

S JS J

2

2

0, ,

0

0

0

tI

I

IJ J I

I

,

,

Q P

q p


Note on J

• There is wide conformance that the canonical symplectic

structure should be

• Not so uniform convention on J

2

1

ni

i

i

dp dq

1

1

1

1

1

1

0, , , ,

0

0 1

1 0

, , , ,

0 1

1 0

0, , , ,

0

n

n

n

n

n

n

Iq q p p J

I

q p q p J

Ip p q q J

I


Definitions Equivalent

• Invariance of fundamental form

• Symplectic matrix definition

1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1

i in n n n

i j ji ik i kj j

j j j jk k

i in n n n n n ni j k ji i

i kj k ji i j k i j k k

in n n nki i

jki j k j kj j

P PQ QdQ dq dp dP dq dp

q p q p

P PQ QdP dQ dq dq dq dp

q q q p

P PQdp dq

p q p

1

in n

j k

i k

Qdp dp

p

0

0

t tt t

t tt t

P PQ P Q PQ Q

I q pq q q qq p

I P P Q QQ P Q P

q p q pp p p p


• If fundamental form invariant CtA, and DtB are symmetric

matrices. Also DtA-BtC is the identity matrix (note the

second term in the form becomes BtC when the dummy

indices are switched), as is its transpose AtD-CtB. This

means the above matrix is J.

• If the above matrix is J, then clearly CtA = AtC and DtB=

BtD, and both DtA-BtC and AtD-CtB (which are transposes

of each other) are the identity matrix. Therefore, the

fundamental form is invariant.

t tt t

t tt t

Q P P Q Q P P Q

q q q q q p q p

Q P P Q Q P P Q

p q p q p p p p


Eigenvalues

• No zero eigenvalues. If an eigenvalue so is

• Eigenvalue equation has real coefficients. Therefore if is

an eigenvalue so is

• General picture

1

1

2

det det det

det det det 1

det det /

t

n

S E S E JS J E

S E E S S

S E S E

1/

*


Strong Stability

• Stable Definition

• Strong Stability Definition

• Theorem: if all 2n eigenvalues of a symplectic

transformation S are distinct and lie on the unit circle in the

complex plane, then S is strongly stable.


Darboux’s Theorem (“Arnold”)

• Theorem (Darboux): Let ω2 be any closed non-degenerate

differential 2-form in a neighborhood of a point x in the

space R2n. Then in some neighborhood of x one can

choose a coordinate system (q1,…,qn,p1,…,pn) such that the

form has “the standard” form

• This theorem allows us to extend to all symplectic

manifolds any assertion of a local character which is

invariant with respect to canonical transformations and is

proven for the standard phase space (R2n, ω2=dp ˄dq)

• For physicists and dynamics: ALL phase spaces have

canonical coordinates where ω2 is given as above

2

1

ni

i

i

dp dq

physics 451/551 theoretical mechanics...theoretical mechanics fall 2018 action-angle variables •...

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