PHY411. PROBLEM SET 8
1. Continuum Lagrangians
R
Consider a continuum system of x with total Lagrangian L = dxL. Suppose
we have a Lagrangian density L(y, ẏ, y 0 , y 00 ) with y 0 = dy/dx and y 00 = d2 y/dx2 .
Show that the Lagrange-Euler equations are
∂L
∂ ∂L
∂ ∂L
∂2
∂L
=
+
− 2
∂y
∂t ∂ ẏ
∂x ∂y 0
∂x ∂y 00
You will need to integrate by parts twice in the integrand of the variation of the
action ∂S. You need to assume nice (zero) boundary conditions for the action.
2. Functional Derivatives
a) Let
Z
F [φ] =
dx
dφ
dx
n
for integer |n| > 1. Show that
" #
n−1
δF [φ]
d dφ
= −n
δφ(y)
dx dx
y
b) For n = 1 show that
δF [φ]
=0
δφ(y)
(assume well behaved zero boundaries for the integral).
c) Let h be a function and
Z
G[φ] =
dx h
dφ
dx
Using the chain rule show that
δG[φ]
d
dh
=−
δφ(y)
dx d(dφ/dx) y
1
2
PHY411. PROBLEM SET 8
d) Consider the functional at the point x
−1
dφ(x)
Fx [φ] =
dx
Show that
δFx [φ]
1
∂
= − dφ(x)
δ(x − y)
δφ(y)
)2 ∂x
(
dx
3. A computation with a non-canonical Poisson bracket
A three-component model to describe shallow water waves with radial and time
dependence and non-canonical coordinates is
1
H(u, v, h) =
h(u2 + v 2 ) + h2 − h
2
and a Poisson bracket
{f, g} = q (∂u f ∂v g − ∂v f ∂u g) + ∂h g∂u f − ∂h f ∂u g
with
(µ + v)
h
Here q corresponds to the potential vorticity and µ is a Coriolis parameter and
∂
∂u = ∂u
.
q=
Show that q is a conserved quantity.
Because H and q are conserved the system reduces to 2D phase space.
Apparently this system can be derived using a Galerkin projection of the nonlinear terms onto the linear modes (see works by Tribbia 1981, Leith 1986, review
by Onno Bokhove).
It may sometimes be convenient to transform a system so that the Hamiltonian
is simpler (or lower order) moving the non-linearities into the Poisson bracket.
4. Sine-Gordon equation in 1 dimension
The Sine-Gordon equation in 1 dimension is
∂ 2u ∂ 2u
− 2 + sin u = 0
∂t2
∂x
PHY411. PROBLEM SET 8
3
Show that the equations of motion arise from a Hamiltonian
!
2
Z ∞
1 2 1 ∂u
π +
− cos u
dx
H[u, π] =
2
2 ∂x
−∞
and a canonical Poisson bracket.
Find soliton solutions by searching for solutions in the form f (x − ct).
You may need to write your solution in terms of the solutions of a different
differential equation (or an elliptic integral).
5. Vortices in 2-dimensions
In two dimensions incompressible flow can be described with a velocity u =
(ux , uy )
ux = ∂y ψ
uy = −∂x ψ
with ψ the stream function. By using the stream function we ensure that ∇·u = 0.
The system can be described with a Hamiltonian
H(x, y) = −ψ
and the Poisson bracket
{f, g} = ∂x f ∂y g − ∂y f ∂x g
The vorticity
ω = ∂x uy − ∂y ux = −(∂x2 + ∂y2 )ψ = −∇2 ψ
Taking the curl of Euler’s equation and rewriting it in terms of vorticity
D
ω = ω · ∇u = 0
Dt
and the last step giving zero follows in two-dimensions as ω is in the z direction.
The vorticity is conserved or advected with the flow.
a) The stream function acts like a potential and satisfies a Poisson equation,
−∇2 ψ = ω. Show that in 2-dimensions the Greens function for a point vortex
at position xi is
Γi
ψ(x) = − log ||x − xi ||
2π
corresponding to a vorticity field
ω(x) = Γi δ(x − xi )
4
PHY411. PROBLEM SET 8
where Γi is a constant. (Show that −∇2 ψ = ω).
b) Show that the associated velocity field is
u(x, y) = ∇ × (ψẑ) =
1
Γi
[(y − yi )x̂ − (x − xi )ŷ]
2π ||x − xi ||2
c) Suppose we have two vortices i, j at xi and xj . The stream function is
Γj
Γi
log ||x − xi || −
log ||x − xj ||
2π
2π
Since the vortices are advected their velocity is equal to that of the surrounding
flow
1
Γj
ẋi (t) =
[(yi − yj )x̂ − (xi − xj )ŷ]
2π ||xi − xj ||2
Γi
1
ẋj (t) = −
[(yi − yj )x̂ − (xi − xj )ŷ]
2π ||xi − xj ||2
ψ=−
Show that the solution is two vortices orbiting in circular orbits a fixed distance
apart. Find the angular rotation rate.
(Using the relative velocity I get an angular rotation rate θ̇ =
distance between them.)
Γi −Γj
2πd2
with d the
d) This system can be modeled with a different Hamiltonian, known as the
Kirchhoff-Routh function
Γi Γj
H(xi , yi , xj , yj ) = −
log ||xi − xj ||
4π
and with the Poisson bracket
1
1 {f, g} =
[∂xi f ∂yi g − ∂yi f ∂xi g] +
∂xj f ∂yj g − ∂yj f ∂xj g
Γi
Γj
Giving equations of motion
ẋi = {xi , H} = Γ−1
i ∂yi H
ẏi = {yi , H} = −Γ−1
i ∂ xi H
−1
ẋj = {xj , H} = Γj ∂yj H
ẏj = {yj , H} = −Γ−1
j ∂xj H
Show that we get the same equations of motion with this new Poisson bracket
and Hamiltonian as we did previously with the Eulerian continuous version.
PHY411. PROBLEM SET 8
5
e) For three or more vortices the evolution of a vorticity field
X
ω(x) =
Γi δ(x − xi )
i
can be modeled with Hamiltonian and Poisson bracket
X Γi Γj
log ||xi − xj ||
H(xi , yi , ....) = −
4π
i6=j
{f, g} =
X 1
[∂xi f ∂yi g − ∂yi f ∂xi g]
Γ
i
i
For three vortices
X
X
J =(
Γi xi )2 + (
Γi yi )2
i
L=
i
X
Γi ||xi ||2
i
and the Hamiltonian itself are conserved quantities (and they are supposed to
be in involution and the system integrable). Check that these are conserved
quantites.
While the full system is infinite dimensional, if we have N vortices we can model
the system with a finite dimensional system with a 2N dimensional phase space.
The passage from Euler to vortex dynamics is described as symplectic reduction.
For 4 or more vortices the dynamics is not necessarily integrable.
Based on notes by Joris Vankerschaver.
6. Transforming to a new Hamiltonian and Poisson Bracket using Burger’s
equation
We start with a Lagrangian density
Z
1
L[x, ∂T x] =
(∂T x)2 dX
2
where X are the Lagrangian coordinates and x the Eulerian ones and the deformation x(X, T ) that transforms between Lagrangian or material coordinates
and the Eulerian ones. Here the material density q(X) in Lagrangian coordinates
X is a constant and we set q(X) = 1. The one dimensional velocity u = ∂T x.
Lagrange’s equations give ∂T T x = 0. This is equivalent to ∂T u = 0 which gives us
u,t + uu,x = 0
6
PHY411. PROBLEM SET 8
and this we recognize as the inviscid Burger’s equation. We construct a momentum
δL
= ∂T x
π(X) =
δ∂T x
and a Hamiltonian
π(X)2
dX
2
Z
H[x, π] =
that has a canonical Poisson bracket with fields x(X), π(X). The density in the
Eulerian coordinates
1
ρ(x) =
J
with Jacobian
J=
∂x
∂X
and momentum density
m(x, T ) = ρu =
∂T x(X, T )
π(X)
=
J(x(X))
J(X)
a) Show that the Hamiltonian as a function of fields ρ(x), m(x) is
Z
1 m(x)2
dx
H[ρ, m] =
2 ρ(x)
(1)
Notice that this is integrated with x rather than X.
b) Consider a functional
Z
F [ρ, m] =
f (ρ(x), m(x))dx
Changing variables
Z
F [ρ, m] =
f (ρ(x(X), m(π(X), ρ(x(X))))JdX
with J = dx/dX = Dx = 1/ρ. Calculate
δF
δx
(2)
PHY411. PROBLEM SET 8
7
Show that
δF
δx
=
=
=
=
∂f 1
∂f π
∂X
+
− ∂X f
∂ρ J ∂m J
∂f
∂f
∂f
∂f
ρ + ∂X
m −
∂X ρ −
∂X m
∂X
∂ρ
∂m
∂ρ
∂m
∂f
∂f
ρ∂X
+ m∂X
∂ρ
∂m
δF
δF
J ρ∂x
+ m∂x
δρ
δm
Notes: The functional
equation 2.
∂f
∂ρ
(3)
δF
δρ
using a functional derivative of
∂f
δF
=
∂m
δm
(4)
can be written as
c) Show that
δF
δπ
=
d) Suppose we also have a functional
Z
G[ρ, m] = g(ρ(x), m(x))dx
Because x, π of X are canonical coordinates
Z
δG
δF
δG
δF
−
{F, G} = dX
δx(X) δπ(X) δx(X) δπ(X)
Show using equations 3 and 4 that the Poisson bracket can be written
Z
δG δF
δF δG
− ρ∂x
{F, G} =
dx ρ∂x
δρ δm
δρ δm
δF
δG
δG
δF
+∂x
m
− ∂x
m
δm
δm
δm
δm
(5)
e) The equations of motion in Eulerian coordinates can be written (in conservation law form)
ρ,t + m,x = 0
2
m
m,t +
= 0
ρ ,x
8
PHY411. PROBLEM SET 8
Using the Hamiltonian as written in equation 1, show that the equations of
motion are consistent with the new Poisson bracket (as in equation 5)
ρ,t = {ρ, H}
m,t = {m, H}
More notes:
δH
m
=
δm
ρ
δH
m2
=− 2
δρ
2ρ
7. Taking new fields ρ, u with u(x(X)) = ∂T x(X) = π(X), show that the equations
of motion for the previous problem are consistent with
Z
ρu2
H[ρ, u] = dx
2
and Poisson bracket
Z
{F, G} =
δF δG
δG δF
dx ∂x
− ∂x
δρ δu
δρ δu
and that the same equations of motion are recovered as in part e of the previous
problem.
8. Matrix Lax Pair for the Liouville Equation
The Liouville equation is
uxt − e2u = 0
(6)
This non-linear equation has a matrix Lax Pair. A Matrix Lax pair is two
matrices X, T that can be a function of a time independent parameter λ. We use
a wavefunction that is a vector ψ. Suppose that our wavefunction satisfies
ψ x = Xψ
ψ t = Tψ
Compute
ψ xt =
=
ψ tx =
=
Xt ψ + Xψ t
Xt ψ + XTψ
Tx ψ + Tψ x
Tx ψ + TXψ
(7)
(8)
PHY411. PROBLEM SET 8
9
Subtracting these two
(Xt − Tx + [X, T]) ψ = 0
A Matrix Lax pair X, T satisfies the zero curvature condition
Xt − Tx + [X, T] = 0
(9)
iff the accompanying differential equation is satisfied.
a) Show that
ux λ
λ −ux
0 −e2u /λ
0
0
X =
T =
are a Lax matrix pair for the Liouville equation. Show that they satisfy the
zero curvature condition (equation 9) if the Liouville equation (equation 6) is
satisifed.
b) Show that
L = ∂x − X
B = T
form an operator Lax pair satisfying ∂t L + [L, B] = 0 when the Liouville
equation is satisfied.
9. Future Problems
Something involving two Poisson brackets?
Symplectic reduction example?
Inverse Scattering Theory problem?