1.
Lecture 9 :
• Phase-space Lagrangian :
Define Lph(q, q̇, p, t) ≡ p·q̇ − H(q, p, t) ,
(1)
= p·[q̇ − u(q, p, t)] + L(q, u(q, p, t), t) .
with Lph(q, q̇, p, t) = L(q, q̇, t) iff q̇ = u(q, p, t).
• Modified Hamiltons principle : Sph stationary iff
δLph ∂Lph
d ∂Lph
∂H
=
−
= −
− ṗ = 0 ,
δq
∂q
dt ∂ q̇
∂q
δLph ∂Lph
∂H
=
= q̇ −
=0.
δp
∂p
∂p
(2)
• Gauge Trans.: Introduce L2ph ≡ L1ph + Ṁ & M = −q·p. Then
L1ph ≡ p·q̇ − H(q, p, t) .
L2ph = −q·ṗ − H(q, p, t) .
produce same equations of motion.•First
(3)
(4)
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• Canonical Transformations : A transformation (q, p) 7→
(Q, P ) that obeys Hamilton’s equations of motion, with new
Hamiltonian K(Q, P ).
0(1)
Lph (Q, Q̇, P , t) ≡ P ·Q̇ − K(Q, P , t) .
(5)
0
• Generating Functions : Sph[q, p] and Sph
[Q, P ] must be the
same, as they describe the same dynamics. Requires existence of
F such that
Lph(q, q̇, p, ṗ, t) = L0ph(Q, Q̇, P , Ṗ , t) +
dF
,
dt
(6)
F can be chosen to be a combination of old (q, p) and new (Q, P )
coordinates
F1(q, Q, t),
F2(q, P , t),
F3(p, Q, t),
F4(p, P , t) .
(7)
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• Generating Functions : (cont.)
0(1)
(1)
Type 1: F1(q, Q, t), Lph = Lph , L0ph = Lph
p=
∂F1
,
∂q
P =−
∂F1
,
∂Q
K=H+
∂F1
∂t
(8)
0(2)
(1)
Type 2: F1(q, P , t), Lph = Lph , L0ph = Lph
p=
∂F2
,
∂q
Q=
∂F2
,
∂P
0(2)
K=H+
∂F2
∂t
(9)
0(1)
Type 3: F1(q, P , t), Lph = Lph , L0ph = Lph
q=−
∂F3
,
∂q
P =−
∂F3
,
∂Q
0(2)
K=H+
∂F3
∂t
(10)
0(2)
Type 4: F1(q, P , t), Lph = Lph , L0ph = Lph
q=−
∂F4
,
∂p
Q=
∂F4
,
∂P
K=H+
∂F4
∂t
(11)
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2.
Identity transformation
The identity transformation is defined by
Q = q,
P =p.
(12)
Type 1 transformations, with generating functions F1(q, Q, t), do not
include the identity because they assume Q̇ is functionally independent of q̇, which is manifestly false.
Consider type 2 generating functions F2(q, P , t):
Q=
∂F2
,
∂P
p=
∂F2
,
∂q
K=H+
∂F2
.
∂t
(13)
Try
F2 = q·P ,
(14)
in eq. (13). This gives Q = ∂F2/∂P = q and p = ∂F2/∂q = P ,
which is the same as eq. (12). That is, eq. (14) generates the identity
transformation.
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3.
Example : Harmonic Oscillator
Problem : Consider the harmonic oscillator with
p2
H=
+ mω 2q 2/2
2m
1
L = m(q̇ 2 − ω 2q 2),
2
(15)
with ω fixed.
Idea : H is the sum of two squares → suggests canonical transformation
f (P )
sin Q
(16)
p = f (P ) cos Q, q =
mω
for which K = H = f 2(P )/(2m).
Solution : Trial
F1 = 12 mωq 2 cot Q
(17)
Substitute into Eqs. (8), providing
p = mωq cot Q,
P = 12 mωq 2 cosec2Q
(18)
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which can be solved, producing
p = (2mωP )1/2 cos Q,
q=
Compare to Eqs. (16), yields f (P ) =
√
2P
mω
1/2
sin Q
(19)
2mωP , and new Hamiltonian
p2
mω 2q 2
K=
+
= ωP
2m
2
(20)
Equations of motion are thus
∂K
=ω
∂P
∂K
=0.
Ṗ = −
∂Q
Q̇ =
(21)
(22)
If = 0, then Q̇ = ω, with solution Q = ωt + α, P = K/ω.
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Interpretation : Solutions for new and old coordinates:
q
√
2H
q = mω2 sin(ωt + α), p = 2mH cos(ωt + α)
Q
= ωt + α,
P = K/ω
(23)
(24)
(25)
Relationship to Quantum mechanics : Replace K ↔ E and P ↔
N ~, with N the number of photons. Hence (Einstein, 1911),
• E = N ~ω, the energy of N photons at frequency ω.
• P = N ~ is a constant of the motion. That is, the number of quanta
is conserved in the system.
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4.
Identity-connected transformations
Suppose we move continuously away from the identity among a family
of canonical transformations parameterized by , say, such that = 0
is the identity. The generating function can be written in the form
F2 = q·P + σ(q, P , t) + O(2) ,
(26)
where the notation O(2) means a term scaling like 2 in the limit
→ 0, and thus negligibly small compared with the O() term.
The term σ is called the infinitesimal generator of the family of
transformations.
Inserting eq. (26) in eq. (13) we have
∂σ(q, P , t)
∂F2
= q+
+ O(2) ,
∂P
∂P
∂F2
∂σ(q, P , t)
= P+
+ O(2) .
p=
∂q
∂q
Q=
(27)
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5.
Infinitesimal canonical transformations
Solve eq. (27) by iteration to give, up to first order in ,
∂σ(q, p, t)
+ O(2) ,
∂p
∂σ(q, p, t)
P (q, p, t, ) = p −
+ O(2) .
∂q
Q(q, p, t, ) = q +
(28)
Also, from eq. (13),
K(Q, P , t, ) = H(q, p, t) +
∂σ(q, p, t)
+ O(2) .
∂t
(29)
This is again a mixed expression in the perturbed and unperturbed
variables so we use eq. (28) to get an explicit expression
∂σ(q, p, t) ∂H ∂σ(q, p, t) ∂H
·
−
·
K(q, p, t, ) = H(q, p, t) +
∂q
∂p
∂p
∂q
∂σ(q, p, t)
(30)
+
+ O(2) .
∂t
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6.
Time evolution
Take = δt. Now consider the choice σ = H in eq. (28). To O(δt),
∂H
δt = q(t) + q̇(t)δt ,
∂p
∂H
P (q, p, t, δt) = p −
δt = p(t) + ṗ(t)δt .
∂q
Q(q, p, t, δt) = q +
i.e. Q(q, p, t, δt) = q(t + δt) ,
P (q, p, t, δt) = p(t + δt) .
(31)
(32)
Hence, the Hamiltonian is the infinitesimal generator for dynamical evolution in time.
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7.
Hamilton-Jacobi Equations
Canonical transformations can be used to provide a general procedure
to solve problems. Two possibilities are:
• If H is conserved, then transform to a new canonical coordinates
that are all cyclic → equations of motion with trivial solutions.
• Seek a canonical transformation from coordinates and momenta
(q, p) at time t to a set of new quantities, which can be initial
values (q0, p0) at t = 0. That is, we want
q = q(q0, p0, t),
p = p(q0, p0, t)
(33)
Procedure : Insist that the transformed Hamiltonian, K = 0.
Then
∂K
∂K
= Q̇ = 0, −
= Ṗ = 0
(34)
∂P
∂Q
K is related to H and generating function via
K = H(q, p, t) +
∂F
=0
∂t
(35)
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Hamilton-Jacobi : (cont.)
Using a type 2 generating function F(q, P, t), then p = ∂F2/∂q, and
so
∂F2
∂F2
H q,
,t +
=0
(36)
∂q
∂t
which is known as the Hamilton-Jacobi equation. The solution of
F2 is often referred to as Hamilton’s principal function, and labeled S.
Equation (36) is a 1st order PDE in n + 1 variables. Suppose there
exists a solution of form
S = S(q1, ..., qn, α1, ..., αn, t)
(37)
where the α1, ..., αn are independent constants of integration. We are
free to choose these to be the new constant momenta, Pi = αi. The F2
transformation equations (9) give
∂S(q, α, t)
∂qi
∂S(q, α, t)
Qi = βi =
∂αi
pi =
(38)
(39)
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Hamilton-Jacobi : (cont.)
The constant βi values can be obtained by the initial conditions, by
computing the RHS of Eq. (39) at t = t0 with known values of qi.
Eq. (39) can be inverted to give q in terms of α, β and t. Differentiating
the RHS of Eq. (38) and substituting gives p in terms of α, β and t.
That is
q = q(α, β, t)
p = p(α, β, t)
(40)
(41)
Hamilton’s principal function is thus the generator of a canonical
transformation to constant coordinates and momenta.
When solving the Hamilton-Jacobi equations, we are at the same
time obtaining a solution to the mechanical problem.
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Hamilton-Jacobi : (cont.)
We also note that
and so
dS
= p · q̇ − H = L
dt
Z
S = Ldt + constant
(42)
(43)
If the Hamiltonian does not depend explicitly on time, Hamilton’s principal function can be written:
S(q, α, t) = W (q, α) − at
(44)
where W (q, α) is Hamilton’s characteristic function. The physical
significance can be understood by writing
∂W
dW
q̇i
=
dt
∂qi
(45)
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Finally, substituting Eq. (44) into (38) gives pi = ∂W/∂qi, and so
Z
dW
= p · q̇ → W = p · q̇dt
(46)
dt
which is an abbreviated action integral.
8.
Lecture 9 : Summary
• Identity transformation
• Example application of generating functions to solve problems.
• Identity connection transformation
• Infinitesimal canonical transformations and time evolution
• Hamilton-Jacobi theory
next lecture : Example use of Hamilton-Jacobi theory.
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