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Theoretical Mechanics Fall 2018
Physics 451/551
Theoretical Mechanics
G. A. Krafft
Old Dominion University
Jefferson Lab
Lecture 11
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
Real Pendulum Phase Space
• Hamiltonian
• Action is the phase space area
2
2cos
2
PH mgL
mL
p
H
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
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
Theoretical Mechanics Fall 2018
• 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
Theoretical Mechanics Fall 2018
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/
*
Theoretical Mechanics Fall 2018
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.
Theoretical Mechanics Fall 2018
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