Physics 311
University of Wisconsin-Madison
Spring 2022
Variational Principles
April 18, 2022
Contents
1 Canonical Transformations
2
2 Hamilton-Jacobi Equation
4
3 Interpretation of Hamilton’s Principle Function as the Classical Action
6
4 Classical Action as the the Phase of the Wavefunction
4.1 Motivation to Set the Classical Action as the Phase . . . . . . . . . . . . . . . . . .
7
8
5 Derivation of the Schrödinger Equation from Hamiltonian Dynamics
9
6 General Definition of a Lagrangian
10
7 Lagrangian Formulation of Special Relativity
10
8 Lagrangian Formalism for a Particle in an Electromagnetic Field
12
1
Physics 311
1
University of Wisconsin-Madison
Spring 2022
Canonical Transformations
The canonical equations of Hamiltonian dynamics are
q̇i =
∂H
∂pi
and ṗi = −
∂H
∂qi
(1)
for a Hamiltonian of the form H(qi , pi , t).
Imagine that we could perform a canonical transformation to a new set of coordinates (Qi , Pi )
and a transformed Hamiltonian K(Qi , Pi ) (which may have a different functional form from H)
that would make solving certain classes of problems easier, and that would still have the same form
of the canonical equations:
Q̇i =
∂K
∂Pi
and Ṗi = −
∂K
∂Qi
(2)
This would allow use preserve all the advantages of Hamiltonian dynamics (e.g., equations of
motion are a system of first-order differential equations, phase space) for a more convenient set of
coordinates.
It turns out that there is a standard mathematical procedure to define such canonical transformations. Begin with the modified Hamilton’s principle
Z
t2
Ldt = 0
δ
(3)
t1
which can be rewritten in terms of the Hamiltonian as
Z
t2
"
#
X
δ
t1
pi q̇i − H(qi , pi , t) dt = 0
(4)
i
Now for the canonical equations to hold for the transformed coordinates, there must be an analogous
relation
Z
t2
"
#
X
δ
t1
Pi Q̇i − K(Qi , Pi , t) dt = 0
(5)
i
Notice that because the modified Hamilton’s principle has zero variation at the endpoints, we could
modify the statement above as
Z
t2
δ
t1
"
X
i
#
dF
Pi Q̇i − K(Qi , Pi , t) +
dt = 0
dt
since
2
(6)
Physics 311
University of Wisconsin-Madison
Z
t2
δ
t1
Spring 2022
dF
dt = δ (F (t2 ) − F (t1 ))
dt
(7)
We know that the modified Hamilton’s principle holds for the Hamiltonian expressed in (qi , pi )
coordinates. If the integrands of the two variational expressions are equal, then the canonical
equations will also hold in the transformed coordinates (Qi , Pi ), that is
X
pi q̇i − H(qi , pi , t) =
i
X
Pi Q̇i − K(Qi , Pi , t) +
i
dF
dt
(8)
In the following, suppress the subscripts and implied summation in our notation for simplicity.
At this point, F is an arbitrary function of phase space coordinates with continuous second derivatives. If F is a function of phase space coordinates, with half of the variables from the old set
of coordinates and half of the variables from the new set of coordinates, then F acts as a bridge
to connect the old and new canonical coordinates. We call F a generating function of the
transformation.
There are four possibilities for the form of the generating function to connect the old and new set
of canonical coordinates:
F = F1 (q, Q, t)
F = F2 (q, P, t) − Qi Pi
F = F3 (p, Q, t) + qi pi
(9)
F = F4 (p, P, t) + qi pi − Qi Pi
We work through the example of F = F2 (q, P, t) − Qi Pi , which is the one used in the HamiltonJacobi formulation of mechanics.
∂F
∂F2
∂F2
∂F2
=
q̇ +
Ṗ +
− Q̇P − QṖ
∂t
∂q
∂P
∂t
(10)
We than have the relation
pq − H = P Q − K +
∂F2
∂F2
∂F2
q̇ +
Ṗ +
− Q̇P − QṖ
∂q
∂P
∂t
(11)
Collect terms
∂F2
p−
∂q
∂F2
∂F2
q̇ = H − K +
+
− Q Ṗ
∂t
∂P
(12)
This yields canonical equations for the new coordinates
pi =
∂F2
∂qi
3
(13)
Physics 311
University of Wisconsin-Madison
Qi =
∂F2
∂Pi
Spring 2022
(14)
and an expression for the transformed Hamiltonian
K =H+
∂F2
∂t
(15)
Similar relations exist for the other functional forms of the generating function F .
Stepping back for a moment, we see that the canonical transformation is a generalization of the
concept of generalized coordinates we encountered in Lagrangian dynamics, but this time allowing even greater flexibility to define coordinate transformations that simultaneously include both
positions and momenta. Furthermore, consider that the number of cyclic (ignorable) coordinates
depends on our choice of coordinates. For example, if we consider a system that is symmetric in
a polar coordinate system, it may be that using polar coordinates will involve a cyclic coordinate,
whereas expressing the problem in terms of cartesian coordinates (x, y) will have no cyclic coordinates. We could therefore attempt to transform to coordinates that would increase the number of
cyclic coordinates and thus yield equations of motion with straightforward solutions. An alternative approach is to transform between coordinates at one time to coordinates at another time, the
strategy that is adopted in Hamilton-Jacobi theory.
2
Hamilton-Jacobi Equation
We consider a specific canonical transformation from the coordinates and momenta (q, p) at time
t to a new set of constant quantities, which may be the 2n initial values at time t = 0, that is
(Q, P ) = (q0 , p0 ). The transformation between these two coordinates is in fact the solution to the
equations of motion in terms of the initial conditions
q = q(q0 , p0 , t)p = p(q0 , p0 , t)
(16)
Q = Q(q, p, t)P
(17)
The transformation is invertible
= P (q, p, t)
Finding such transformations amounts to solving the problem.
In this case, we seek to transform from canonical variables (qi , pi ) to a new set of variables (Q, P )
that follow the canonical equations
Q̇i =
∂K
∂Pi
and Ṗi = −
∂K
∂Qi
(18)
where the new coordinates are constant, that is
Q̇i = 0
and Ṗi = 0
4
(19)
Physics 311
University of Wisconsin-Madison
Spring 2022
We can ensure that the new coordinates are constant by setting the transformed Hamiltonian to
zero, K = 0:
Q̇i =
∂K
=0
∂Pi
and Ṗi = −
∂K
=0
∂Qi
(20)
This further implies that
0=H+
∂F
∂t
(21)
At this point, we need to select the variables of the generating function. Choose a generating
function of the form F = F2 (q, P, t) − Qi Pi For historical reasons that will become apparent in a
monent, it is customary to denote this generating function as S. With this notation, we have
pi =
∂S
∂qi
(22)
Qi =
∂S
∂Pi
(23)
K =H+
∂S
∂t
(24)
∂S
=0
∂t
(25)
Since K = 0, we can write
H(qi , pi , t) +
and substituting for pi we arrive at the Hamilton-Jacobi equation
∂S
∂S
H qi ,
=0
,t +
∂qi
∂t
(26)
Given the functional form of H, one can solve for S. We have now reduced the problem of solving
the entire system of equations of motion to solving a single partial differential equation for the
function S. This a significant accomplishment from a theory perspective, but in practice, the
partial differential equation for S is often very difficult to solve and problems are usually more
easily solved by other means. Still, the Hamilton-Jacobi equation has important applications,
including a direct connection to the Schrödinger equation in quantum mechanics.
Historically, S was called “Hamilton’s principle function”. Nowadays, we usually call S the action,
and it does in fact correspond to the action we encountered in Lagrangian dynamics.
5
Physics 311
3
University of Wisconsin-Madison
Spring 2022
Interpretation of Hamilton’s Principle Function as the Classical
Action
We treated Hamilton’s principle function S as a generating function of a canonical transformation
of the form S(qi , Pi , t). Compute the time derivative
X ∂S dqi X ∂S dPi ∂S
dS
=
+
+
dt
∂qi dt
∂Pi dt
∂t
(27)
∂S
= pi
∂qi
(28)
dPi
=0
dt
(29)
but
from the canonical transformation and
by construction because we were were specifically using a canonical equation for which the new
variables would be constants representing the initial conditions at time t = 0. We then have
X
dS
∂S
=
pi q̇i +
dt
∂t
(30)
but the Hamiltonian is
H=
X
pi q̇i − L
(31)
∂S
dS
=H+L+
dt
∂t
(32)
so
and the Hamilton-Jacobi equation states
H+
∂S
=0
∂t
(33)
so
dS
=L
dt
(34)
Ldt + constant
(35)
We find that
Z
S=
6
Physics 311
University of Wisconsin-Madison
Spring 2022
and thus the generating function differs from the classical action by at most an additive constant.
Since we are interested primarily in the variation of the action, we see that Hamilton’s principle
function can be practically identified with the classical action.
4
Classical Action as the the Phase of the Wavefunction
Guess that the wavefunction is of the form
iS
(36)
ψ = ae h̄
with the classical action S determining the phase of the wavefunction.
Then in the one-dimensional case with x the generalized coordinate
iS
∂ψ
= ae h̄
∂x
i ∂S
h̄ ∂x
=ψ
i ∂S
h̄ ∂x
(37)
Rearrange
∂S
ih̄ ∂ψ
=−
∂x
ψ ∂x
(38)
∂ψ
∂S
= −ih̄
∂x
∂x
(39)
∂S
=p
∂x
(40)
Notice that we can further rearrange as
ψ
Now
where we recognize S as the generating function of the canonical transformation. We have
− ih̄
∂ψ
= pψ
∂x
(41)
This result motivates the momentum operator in quantum mechanics, p̂
p̂ = −ih̄
∂
∂x
(42)
The momentum operator acting on a momentum eigenstate yields an eigenvalue (the momentum)
multiplied by the eigenstate.
p̂ψ = pψ
7
(43)
Physics 311
4.1
University of Wisconsin-Madison
Spring 2022
Motivation to Set the Classical Action as the Phase
Consider an analogy with optics. Light waves obey a wave equation of the form
∇2 φ −
n2 ∂ 2 φ
=0
c2 ∂t2
(44)
with solutions of the form
φ = φ0 eik(nz−ct)
(45)
for a wave propagating in the z direction, for instance. The optical wavefront is a surface of constant
phase.
Now, what about a matter wave? If the Hamiltonian has no explicit dependence on time, the
Hamilton-Jacobi equation becomes
∂S
∂S
+ H qi ,
=0
∂t
∂qi
(46)
The time variable can then be separated by assuming a solution for S of the form
S(q, P, t) = W (q, P ) − αt
(47)
where W is independent of time. The function W , known as Hamilton’s characteristic function,
generates a canonical equation in which all of the new coordinates are cyclic. Substituting this
solution yields
−α−H=0
(48)
and we find that α = H, a constant that usually corresponds to the total mechanical energy
α = H = E in conservative system. If the Hamiltonian is indeed a constant of motion and is
identical with the total energy, then we can write
S(q, P, t) = W (q, P ) − Et
(49)
Since the characteristic function is independent of time, surfaces of constant W in configuration
space have fixed locations. Meanwhile, the action evolves in time as
dS
= pi q̇i − H = L
dt
(50)
We see that a surface of constant S in phase space propagates in the same way that an ordinary
wavefront propagates in configuration space.
Notice that the action and Planck’s constant have the same units, and since the phase must be
dimensionless, we might guess the phase is Sh̄ .
8
Physics 311
5
University of Wisconsin-Madison
Spring 2022
Derivation of the Schrödinger Equation from Hamiltonian Dynamics
The Hamilton-Jacobi equation
H(qi ,
∂S
∂S
, t) +
=0
∂qi
∂t
(51)
p2i
+U
2m
(52)
Consider Hamiltonian of the form
H=
In the one-dimensional case with x as the generalized coordinate, the Hamilton-Jacobi equation
reduces to
1
∂S
+
∂t
2m
∂S
∂x
2
+U =0
(53)
Now the Hamilton-Jacobi equation states
∂S
∂S
=E+
=0
∂t
∂t
H+
(54)
where we recognize the Hamiltonian as the being equal to the total mechanical energy in a conservative system with scleronomic coordinates. Considering the complex nature of the wavefunction,
use the complex conjugate
1
−E+
2m
∂S
∂x
∗ ∂S
∂x
+U =0
(55)
and now we substitute our result from above
1
−E+
2m
!
ih̄
−
∗
ψ ∗ ∂ψ
∂x
ih̄
!
ψ ∂ψ
∂x
+U =0
(56)
Rearrange
h̄2
(U − E)ψ ψ +
2m
∗
∂ψ ∗
∂x
∂ψ
∂x
(57)
Call the whole left hand side of the equation M . We are about to use the calculus of variations
again in perhaps an unexpected way. Consider the variation
Z
δ
M dx = 0
9
(58)
Physics 311
University of Wisconsin-Madison
Spring 2022
will result in the Euler-Lagrange equation


∂  ∂M 
∂M
=0
−
∂ψ ∗ ∂x ∂ ∂ψ∗
(59)
∂ h̄2 ∂ψ
=0
(U − E)ψ −
∂x 2m ∂x
(60)
∂x
from which we find
or
(U − E)ψ −
h̄2 ∂ 2 ψ
=0
2m ∂x2
(61)
This is the Schrödinger equation.
6
General Definition of a Lagrangian
For a given mechanical system with generalized coordinates qi , a Lagrangian is a function L(qi , q̇i )
of the coordinates and velocities such that the correct equations of motion for the system are the
Euler-Lagrange equations
∂L
d ∂L
=
∂qi
dt ∂ q̇i
7
(62)
Lagrangian Formulation of Special Relativity
The Lagrangian is classical mechanics is
L=T −U
(63)
1
T = mv 2
2
(64)
T = (γ − 1)mc2
(65)
with kinetic energy
In special relativity, the kinetic energy is
where the Lorentz factor is defined as
10
Physics 311
University of Wisconsin-Madison
1
γ=p
1 − β2
Spring 2022
(66)
with
β2 =
v2
c2
(67)
We claim that the Lagrangian for special relativity is
L = −mc2
p
1 − β2 − U
(68)
Euler-Lagrange equation
d
dt
∂L
∂vi
=
∂L
∂xi
(69)
We find that the generalized momentum is the relativistic momentum
∂L
mvi
=p
= γmvi = pi
∂vi
1 − β2
(70)
We then arrive at a version of Newton’s Second Law that is appropriate for relativistic motion
d
∂U
(pi ) = −
= Fi
dt
∂xi
(71)
Next, consider the Hamiltonian
H=
X
pi q̇i − L
(72)
i
Without loss of generality, we can choose one of the axes to be oriented in the direction of motion
so that
p
mv 2
+ mc2 1 − β 2 + U
H= p i
1 − β2
(73)
We find that the total mechanical energy is exactly what we would expect from special relativity
mc2
H= p
+ U = T + U + mc2 = E
2
1−β
11
(74)
Physics 311
8
University of Wisconsin-Madison
Spring 2022
Lagrangian Formalism for a Particle in an Electromagnetic Field
Consider a particle in an electric and magnetic field that are determined by the scalar potential
~ r, t). The electric field is
V (~r, t) and vector potential A(~
~
~ = −∇V
~ − ∂A
E
∂t
(75)
~ = −∇
~ ×A
~
B
(76)
~ + ~r˙ × B
~
m~r¨ = q E
(77)
1 2
~
L(~r, ~r˙, t) = m~r˙ − q(V − ~r˙ · A)
2
(78)
and the magnetic field is
Lorentz force law
We claim that the Lagrangian is
and show that this form of the Lagrangian yields the correct equations of motion when we apply
the Euler-Lagrange equations.
Expand the Lagrangian
1
L = m ẋ2 + ẏ 2 + ż 2 − q (V − ẋAx − ẏAy − żAz )
2
(79)
Examine the Euler-Lagrange equation for the x coordinate
∂L
d
=
∂x
dt
∂L
= −q
∂x
∂L
∂ ẋ
∂Ay
∂V
∂Ax
∂Az
− ẋ
− ẏ
− ż
∂x
∂x
∂x
∂x
(80)
∂L
= mẋ + qAx
∂ ẋ
(81)
(82)
Recall that Ax = Ax (x, y, z, t) and apply the chain rule:
d
dt
∂L
∂ ẋ
Ax
Ax
Ax ∂Ax
= mẍ + q ẋ
+ ẏ
+ ż
+
∂x
∂y
∂z
∂t
Combine
12
(83)
Physics 311
University of Wisconsin-Madison
mẍ = −q
∂Ax
∂V
+
∂x
∂t
+ q ẏ
∂Ay
∂Ax
−
∂x
∂y
Spring 2022
− q ż
∂Ax ∂Az
−
∂z
∂x
mẍ = q (Ex + ẏBz − żBy )
(84)
(85)
We can recognize this as the x component of the Lorentz force law. The same logic applies to the
y and z coordinates.
Consider the generalized momentum. First, the x coordinate:
px =
∂L
= mẋ + qAx
∂ ẋ
(86)
The y and z components work the same way, so we find
~
p~ (generalized momentum) = m~v + q A
(87)
~
The generalized momentum is the sum of the mechanical momentum and a new term q A.
~ depends on the x coordinate.
Imagine that we have a single particle in a field where neither V not A
Then x would not appear in the Lagrangian and would be cyclic, so px is a conserved quantity.
px = mẋ + qAx = constant
(88)
It can be shown from classical electrodynamics that under these conditions, that qAx is exactly the
x component of the electromagnetic linear momentum of the field associated with the charge q. In
essence, we have found that the electromagnetic field itself can carry momentum.
In special relativity context, we could write the Lagrangian as
L = −mc2
p
~
1 − β 2 − q(V − ~r˙ · A)
(89)
and we would find the same result that the linear momentum includes both the mechanical momentum and the momentum of the electromagnetic field
pi = γmvi + qAi
13
(90)