Tentative Title: Nonlinear Waves and

Integrability I: Derivation of Basic Integrable and

non-Integrable Equations
Evgeny A. Kuznetsov
Pavel M. Lushnikov
Vladimir E. Zakharov
Lebedev Insitute of the russian academy of sciences, moscow, russia
E-mail address: kuznetso@itp.ac.ru
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA
E-mail address: plushnik@math.unm.edu
Department of Mathematics, University of Arizona, PO Box 210089,
Tucson, Arizona, 85721, USA
E-mail address: zakharov@math.arizona.edu

2010 Mathematics Subject Classification. Primary 54C40, 14E20;

Secondary 46E25, 20C20 ???
Key words and phrases. amsbook, AMS-LATEX

Evgeny A. Kuznetsov was supported in part by NSF Grant #000000.

Pavel M. Lushnikov was supported in part by NSF Grant #000000.
Vladimir E. Zakharov was supported in part by NSF Grant #000000.

Contents
Part 1.

Hamiltonian Formalism

Chapter 1. Finite Dimensional Canonical Hamiltonian Systems

1.1. Canonical Hamiltonian mechanics
1.2. Poisson bracket and action-angle variables
1.3. Canonical transformations and generating functions
1.4. Canonical transformation through Poisson bracket
1.5. Lagrangian mechanics
1.6. Equivalence of the Lagrangian and Hamiltonian mechanics
1.7. Holonomic constraints and the Lagrangian mechanics on manifolds
1.8. Oscillations
Exercise 1.
1.9. Harmonic oscillator and complex variables
1.10. Complex variables in weakly nonlinear case
1.11. Nonlinear oscillator and the generation of multiple harmonics
Exercise 2.
Exercise 3.
Exercise 4.
Exercise 5.
1.12. Canonical transformations
1.13. Generalization of the canonical Hamilton equations. Symplectic
structure. Poisson mechanics.
1.14. Symplectic leaves. Example of free motion of rigid body

3
3
5
6
9
10
11
12
14
16
17
19
20
22
23
23
25
25

Bibliography

27

Index

29

iii

25
25

Part 1

Hamiltonian Formalism

CHAPTER 1

Finite Dimensional Canonical Hamiltonian

Systems
We start by recalling basic facts from classical mechanics. More details can be
found in the books on classical mechanics, see e.g. [LL89, Arn89]
1.1. Canonical Hamiltonian mechanics
Canonical Hamilton equations
dqj
H
=
,
dt
pj
(1.1)
H
dpj
=
,
dt
qj

j = 1, . . . , N

are the system of the even number 2N (N is the positive integer) of ordinary
differential equations (ODE). That system is fully defined by the Hamiltonian H
which is the given smooth function H = H(p1 , . . . , pN , q1 , . . . qN , t) of independent
variables p := (p1 . . . , pN ), q := (q1 . . . , qN ) and time t. Here all components of p,
q and as well as H and t are real numbers R. The variables p and q are usually
called by the generalized momenta and coordinates, respectively. In most cases
below H does not depends on t explicitly (unless we explicitly specify the opposite)
and then the Hamiltonian is the constant of motion as follows from the dynamic
PN
H
H

equations (1.1): dH
j=1 p j pj + qj qj + H = 0, where here and below we use
dt =
the notation f := f
t for any function f (t). The set of all possible values of p and
q is called by a phase space P . The canonical Hamilton equations together with
the independent variables q and p and the phase space P form the equations of the
Hamiltonian mechanics.
For the following particular form of the Hamiltonian
(1.2)

H=

N
X
pi 2
+ U (q)
2mi
i=1

we recover from (1.1) the Newton equations for the motion of the particles with
the coordinates q, the momenta p and the masses mi in the potential U (q):
pi = mi qi , i = 1, . . . N, p = U
q , where here and below we use the notation



q :=
q1 , . . . , qN . For N/D particles in Ddimensional space with coordinates r1 , r2 , . . . , rN/D RD we set q = (r1 , r2 , . . . , rN/D ) RN (masses mi are
split in that case in N/D groups). Each component of q and p can take the arbitrary real values so the phase space is RN RN = R2N , the vector space of 2N real
N
P
pi 2
numbers. In the equation (1.2), the terms
2mi and U (q) are called by the kinetic
i=1

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

Figure 1. A schematic of planar pendulum.???

energy and the potential energy, respectively. Thus the Newton equations of the
potential motion of particles is the particular case of the Hamiltonian mechanics
with the Hamiltonian (1.2) and the phase space R2N .
The system of harmonic oscillators is one of the simplest nontrivial example of
the Hamiltonian (1.2) with the quadratic potential
(1.3)

U (q) =

N
X
mj j2
j=1

qj 2 ,

which gives from (1.2) the harmonic oscillator Hamiltonian

!
N
X
mj j2 2
pj 2
(1.4)
H=
+
qj ,
2mj
2
i=1
which is quadratic both in q and p. Then the dynamic equations (1.1) turns into
the system of linear ODEs with constant coefficients which are decoupled for each
pair of canonical variables (pj , qj ), j = 1, . . . , N . The general solution of that
system is
(1.5)

pj = mj j aj cos (j t + j ),
qj = aj sin (j t + j ),

j = 1, . . . , N,

where a = (a1 , . . . , aN ) and = (1 , . . . , N ) are 2N arbitrary real constants.

These constants are uniquely determined by the initial value problem p(t0 ) = p0
and q(t0 ) = q0 for the system (1.1) and (1.4).
The phase space for the harmonics oscillator is R2N . Generally, the phase space
is a smooth manifold (see Appendix ?? for the definition). Consider the ideal planar
pendulum which is the points mass m in the gravitational field g attached by a rigid
rod of zero mass and length L to the point O such that it can rotate freely at any
angle around O as shown in Figure 1. It is convenient to define the generalized
coordinate as the angle of rotation in counterclockwise direction with respect to
the negative direction of the vertical axis as shown in Figure 1. The Hamiltonian is
2 2
the sum of the kinetic energy m
2 L and the potential energy mgL cos giving
(1.6)

H=

p2
mgL cos ,
2mL2

Equations (1.1) and (1.6)

where we defined the canonical momentum as p = mL2 .
can be solved in elliptic functions. The qualitative behaviour of the system can
be however extracted from the conservation of the Hamiltonian H: if H < mgL
then even for p = 0 the angle cannot reach the value = and the pendilum
oscillates nether passing through the vertical position = ; if H > mgL then
the pendulum rotates nonstop either clockwise of counterclockwise depending on
t=0 . Values of |p(t)| can be arbitrary large
the sign the initial angular velocity |
(depending on initial conditions) while spans between and because of the
periodicity in . Thus the phase space of the planar pendulum is the cylinder
[, ) R.

1.2. POISSON BRACKET AND ACTION-ANGLE VARIABLES

1.2. Poisson bracket and action-angle variables

It is often convenient to formulate the Hamiltonian equations (1.1) through
a Poisson bracket. Consider the functions F (p, q) and G(p, q) assumed to be
infinitely differentiable in both arguments. The Poisson bracket {., .} in canonical
variables p and q is defined in the phase space P as

N 
X
F G
F G
{F, G} =

.
(1.7)
qj pj
pj qj
j=1
The Hamiltonian equations (1.1) can be equivalently expressed through the
Poisson brackets between p, q and the Hamiltonian H as follows
dq
= {q, H},
dt
dp
= {p, H}.
dt

(1.8)

Using the definition (1.7) we immediate obtain that {q, H} = H

p and {p, H} =
H
q thus recovering (1.1) from (1.8).
It follows from the definition (1.7) that the Poisson bracket is antisymmetric
{F, G} = {G, F }

(1.9)
and satisfy a Jacobi identity
(1.10)

{F, {G, H}} + {G, {H, F }} + {H, {F, G}} = 0

for any infinitely differentiable functions F (p, q), G(p, q) and H(p, q).
Time-derivative of any function F (p, q) with p(t) and q(t) being the solution
of (1.1) is conveniently represented through the Poisson bracket as follows
(1.11)

F
F
H F
H F
dF
= p
+ q
=

= {F, H}.
dt
p
q
p p
q q

A function F is called a constant of motion if F is constant along the solution of

(1.1): dF
dt = 0, which implies by (1.11) that
(1.12)

{F, H} = 0.

Two functions F and G are called to be in involution if {F, G} = 0. Thus

(1.12) implies that any constant of motion is in involution with the Hamiltonian. If
there are N functionally independent integrals of motion Fj , j = 1, . . . , N , which
are all involution, i.e. {Fi , Fj } = 0 for any i, j = 1, . . . , N then the Hamiltonian
system is called Liouville integrable (also called by completely integrable system in
Liouville sense). Here by functional independence of Fj we mean that a set of 2N dimensional gradients (p , q )Fj , j = 1, . . . , N spans an N -dimensional subspace
of phase space almost everywhere except on sets of zero measure. In other words,
vectors (p , q )Fj , j = 1, . . . , N are linearly independent almost everywhere in the
phase space. Liouville-Arnold theorem [Arn89] proves for any Liouville integrable
system that if the level set Mf = {(p, q) : Fj = fj , fj R, j = 1, . . . , N, } is
compact then it is diffeomorphic (can be transformed into by a smooth invertible
transformation) to N -dimensional torus T = {1 , . . . , N }, where j R modulo
2.

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

Liouville-Arnold theorem also proves that exists a neighborhood of Mf P

where it is possible to find action-angle variables (I, ) such that
dI
= 0,
dt
(1.13)
d
= I H(I) := (I).
dt
Here we define the action coordinate I RN and the angle coordinate TN in
such a way that the Hamiltonian depends only on the action
(1.14)

H = H(I)

but not on . The action is respectively the function of F = (F1 , . . . , Fj ): I = I(F).

Equations (1.13) are simple to integrate which gives
I(t) = I(0) = const,

(1.15)

(t) = (0) + (I)t.

Thus the solution of Liouville integrable system is quasiperiodic with the rotation
of each angular coordinate j at the angular frequency j , j = 1, . . . , N .
For N = 1 any Hamiltonian system (1.1) is Lioville integrable because we use
the Hamiltonian as the integral of motion H = F1 . E.g., for the harmonic oscillator
2
p2
2
+ m
I = H, where H = 2m
2 q .
1.3. Canonical transformations and generating functions
The differential equations (1.13) in action-angle variables (I, ) is the particular
case of the Hamiltonian equations (1.1) with I being the generalized momentum
and being the generalized coordinate. Thus the change of variables from the
original variables (p, q) into action-angle variables save the general form (1.1) of
the Hamiltonian equations. In this Section we study the general transformation
of the Hamiltonian equations from the original variables (p, q) into new variables
variables (P, Q) such that

dQj
H
=
,
dt
Pj
(1.16)

H
dPj
=
, j = 1...N
dt
Qj
with P RN and Q RN being the new generalized momentum and generalized

coordinate, respectively. Here H(P,

Q, t) is the Hamiltonian in new variables. For
generality in this section we allow the explicit time dependence of the Hamiltonian:
H(p, q, t).
We start by deriving the Hamiltonian equations (1.1) from the principle of least
action (or more accurately is sometimes called by the principle of stationary action
as well as it is known as the Hamiltons principle) for the action S defined as the
integral
Zt2

S = [p(t) q(t)
(1.17)
H (p(t), q(t), t)] dt,
t1

between two fixed times t1 and t2 . The functional S here is understood as the line
integral along a curve in 2N + 1-dimensional real vector space (p, q, t) R2N +1 .

1.3. CANONICAL TRANSFORMATIONS AND GENERATING FUNCTIONS

The principle of the least action states that the motion of the system between the
specified points q1 := q(t1 ) and q2 := q(t2 ) is determined by the the stationary
value (extremal) of the action S. Assume that the functions p(t) and q(t) realize
such stationary value. Then it means that the replacement of p(t) and q(t) in
(1.32) by their independent infinitesimal variations in the form p(t) + p(t) and
q(t) + q(t) with fixed generalized coordinates at initial and finite times,
(1.18)

q(t1 ) = q(t2 ) = 0

do not change the value of S in the linear order of p(t) and q(t). It is assumed
here that p(t), q(t), p(t) and q(t) are smooth functions at t1 t t2 to ensure
the differentiability of S. One can assume for example, that these functions are
infinitely differentiable. The variation of S is given by
Zt2
H (p + p, q + q, t)] dt
[(p + p) (q + q)

S =
t1

Zt2

Zt2 
[p q H (q, p, t)] dt =

t1

(1.19)

p q + p q


H
H
q
p dt
q
p

t1

+ h.o.t.,

where h.o.t. means the higher order terms (quadratic and above) in q and p.
Below unless otherwise specified, we neglect h.o.t. in S, i.e. by S we assume the
first variation (only linear in q and p contributions). Also the scalar products
PN H
H
j=1 qj qj and similar expressions are implied here and below. The
q q =
assumption of the stationary value of S on p(t) and q(t) implies that S = 0. Then
integrating (1.34) by part for the term with q = dq
dt we obtain that
(1.20)

S = p

t=t
q|t=t21

Zt2 
+
t1

H
q
p

Zt2 


p dt +

H
q


q dt = 0.

t1

Taking into account the condition (1.33) we conclude that the integral in (1.35)
vanishes for any independent variations p and q. It means that the expressions
in brackets at each integrand in (1.35) also vanish resulting in the Hamiltonian
equations (1.1). Thus the Hamiltonian mechanics mechanics can be considered as
the reformulation of the classical mechanics using the principle of least action.
We note that the above derivation of the Hamiltonian equations (1.1) from the
variation of S needs that the motion of the mechanical system to be the stationary
value of S but does not actually require S to be the minimum. In most cases
however, for short pieces of the mechanical system trajectories, S is the minimum.
It justifies the historical use of the principle of least action terminology. Although
more accurately it should be called by the principle of stationary action. Later
in the book we additionally discuss the variational principle by introducing the
Lagrangian in Section???
Now we return to the transformation between variables (p, q) which satisfy
(1.1) and (P, Q) which satisfy (1.16). Because of (1.16), new variables must also

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

satisfy the Hamiltons principle S = 0 with

(1.21)

S =

Zt2 h

i
H
(P, Q, t) dt,
PQ

t1

Both S = 0 with (1.17) and S = 0 (1.21) are simultaneously satisfied if the

integrands of (1.17) and S = 0 (1.21) are different by the total time derivative of
any smooth function F as follows
(1.22)

H
+ F
p q H = P Q

2
because the difference in S and S is reduced to the constant term F |t=t
t=t1 which
vanishes under variation.
We consider the transformation

(1.23)

P = P(p, q, t)

and Q = Q(p, q, t)

from the old variables (p, q) to the new variables (P, Q), were p, q, P and Q RN .
The transformation (1.23) is called the canonical transformation if both (1.1) and
(1.16) are simultaneously satisfied and there exists a function F such that (1.22) is
valid. We define F in one of four possible forms as follows:
(1.24a)

F = F1 (q, Q, t),

(1.24b)

F = F2 (P, q, t) P Q,

(1.24c)

F = F3 (p, Q, t) + p q,

(1.24d)

F = F4 (p, P, t) + p q P Q.

Each function Fj , j = 1, . . . , 4 is called the generating function of the canonical

transformation. We call these generating functions as type 1, 2, 3 and 4, respectively. Thus the equations (1.24a)-(1.24d) allow the generating function to be the
function of any pair of the old and new canonical variables and time. Four different
types of canonical transformations are obtained from these four forms of the generating function. Taking the first form F = F1 (1.24a) and substituting the total
time derivative of F1 into (1.22) by F1 result in
(1.25)

H
H
+ F1 = P Q
+
p q H = P Q

F1
F1
F1
q +
Q+
.
q
Q
t

We assume the the old coordinate q and the new coordinate Q are separately independent. Then the equation (1.25) is identically valid if the expressions multiplying
are both identically zero as well as the sum of the remaining terms
both q and Q
is zero which give the expressions for the old and new generalized momenta as well
as the relation between the old and new Hamiltonians as follows
F1
F1
= H + F1 .
p=
, P=
, H
(1.26)
q
Q
t
We can define the transformation (p, q) (P, Q) from (1.26) as follows. The
first equation in (1.26) defines p as the function of q, Q and t. We assume that
this function can be inverted to give Q as the function of p, q and t, thus giving
the second part of the transformation (1.23). Then Q(p, q, t) can be substituted
into the second equation in (1.26) resulting in P(p, q, t), i.e. the first part of the
transformation (1.23). After that using the inverse of the transformation (1.23) in

1.4. CANONICAL TRANSFORMATION THROUGH POISSON BRACKET

in terms of
the third equation in (1.26) we immediately the new Hamiltonian H
the the new canonical variables (P, Q).
In a similar way, taking F from (1.24b),(1.24c) and (1.24d), respectively, and
substituting the total time derivative of F of into (1.22) result in

(1.27a)
(1.27b)
(1.27c)

F2
F2
= H + F2 ,
, Q=
, H
q
P
t
F3
F3
= H + F3 ,
q=
, P=
, H
p
Q
t
F4
F
F4
4
=H+
, Q=
, H
.
q=
p
P
t
p=

The last expression in each of the equations (1.26), (1.27a), (1.27b) and (1.27c)
in terms of the new canonical variables
give the formula for the new Hamiltonian H
(P, Q). We also note from these equations that if the generating function F does
In that case to obtain the new Hamiltonian
not depend on t explicitly then H = H.

H it is sufficient to express p and q in the old Hamiltonian H through the new

variables P and Q.
The canonical transform does not have to be precisely one of four types (1.24a).
Instead it could be a mixture of all four types. As a simple example consider the
generating function of the mixed type
(1.28)

F (p(1) , Q(1) , P(2) , q(2) ),

where p(1) = (p1 , . . . pN1 ), Q(1) = (Q1 , . . . QN1 ), P(2) = (PN1 +1 , . . . PN ) and q(2) =
(qN1 +1 , . . . pN ), 1 N1 N 1, i.e. it is of the type 3 for first N1 components of
the canonical momenta and coordinates and it is of the type 2 for the other N N1
components with F in the equation (1.22) given by
(1.29)

F = F (p(1) , Q(1) , P(2) , q(2) ) + p(1) q(1) P(2) Q(2) .

The distinction between canonical coordinates and canonical momenta is essentially lost under the canonical transformation. This can be seen if we perform
a simple canonical transformation P = q and Q = p which interchanges the
canonical coordinates and canonical momenta. That canonical transformation is
of the type 1 with F = F1 = q Q. It suggest to call canonical coordinates and
canonical momenta as canonically conjugated quantities.
1.4. Canonical transformation through Poisson bracket
The Poisson bracket is invariant with respect to the canonical transformation,
i.e. the Poisson bracket is canonical invariant. To show that we define the Poisson
bracket with respect to the canonical variables Q and P as follows

N 
X
F G
F G
{F, G}Q,P =

.
(1.30)
Qj Pj
Qj Qj
j=1
Then (1.7) defines {F, G}q,p . Substitution of (1.23) results in
Relation through the Poisson brackets.
?

10

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

1.5. Lagrangian mechanics

The canonical Hamiltonian system (1.1) has the equivalent form called by the
Lagrange equations (also sometimes called by the Euler-Lagrange equations or the
Lagrange equations of the second kind)


d L
L
(1.31)

=0
dt q
q
which is the system of N second order (in time) ODEs. That system is fully defined
t) of independent
by the Lagrangian L which is the given function L = L(q, q,
variables t, q RN and q RN (q is used as the second independent variable
in contrast to the Hamiltonian system (1.1), where p is the second independent
variable). The Lagrange equations together with the independent variables q and
q form the Lagrangian mechanics.
The Lagrange equations follows from the principle of least action (or more
accurately as the principle of stationary action as well as it is also called by the
Hamiltons principle) for the action S defined as
Zt2
(1.32)

L (q(t), q(t),
t) dt

S=
t1

between two fixed times t1 and t2 . The functional S here is understood as the
line integral along a curve in N + 1-dimensional real vector space (q, t) RN +1 .
The principle of the least action states that the motion of the system between the
specified points q1 := q(t1 ) and q2 := q(t2 ) is determined by the the stationary
value (extremal) of the action S. Assume that the function q(t) realizes such stationary value. Then it means that the replacement q(t) in (1.32) by its infinitesimal
variation in the form q(t) + q(t) with
(1.33)

q(t1 ) = q(t2 ) = 0

does not change the value of S in the linear order of q(t). It is assumed here
that both q(t) and q(t) are smooth functions at t1 t t2 to ensure the differentiability of S. One can assume for example, that these functions are infinitely
differentiable. The variation of S is given by
Zt2

Zt2
t) dt
L (q + q, q + q,

S =
t1

Zt2 
t) dt =
L (q, q,

t1


L
L
q +
q dt
q
q

t1

(1.34)

+h.o.t.,

where h.o.t. means the higher order terms (quadratic and above) in q and q.
Below unless otherwise specified, we neglect h.o.t. in S, i.e. by S we assume the
first variation (only linear in q and q contribution). Also the scalar products
PN L
L
j=1 qj qj and similar expressions are implied here and below. The asq q =
sumption of the stationary value of S on q(t) implies that S = 0. Then integrating
(1.34) by part for the term with q = dq
dt we obtain that
(1.35)

t=t

Zt2 
L  2
d L
L
S =
q
+

q dt = 0.
q t=t1
q dt q
t1

1.6. EQUIVALENCE OF THE LAGRANGIAN AND HAMILTONIAN MECHANICS

11

Taking into account the condition (1.33) we conclude that the integral in (1.35)
vanishes for any q. It means that the integrand in (1.35) also vanishes resulting in
the Lagrange equations (1.31). Thus the Lagrangian mechanics can be considered
as the reformulation of the classical mechanics using the principle of least action.
We note that the above derivation of the Lagrange equations (1.31) needs that
the motion of the mechanical system to be the stationary value of S but does not
actually require S to be the minimum. In most cases however, for short pieces of
the mechanical system trajectories, S is the minimum. It justifies the historical use
of the principle of least action terminology. Although more accurately it should be
called by the principle of stationary action.
1.6. Equivalence of the Lagrangian and Hamiltonian mechanics
We introduce the generalized momentum p in the Lagrangian mechanics as
follows
L
(1.36)
.
p=
q
Then the Lagrange equations (1.31) can be rewritten as the system of two sets of
equations
L
,
q
L
p=
.
q
p =

(1.37)

We define the relation between the Hamiltonian H(p, q, t) and the Lagrangian
t) describing the same mechanical system as follows
L(q, q,
(1.38)

t).
H(p, q, t) = pq L(q, q,

The equation (1.38) together with (1.38) define the Legendre transform from the
independent variables p, q and t to the independent variables q, q and t. The exact
differential dH of H is given by
(1.39)

dH =

H
H
H
dp +
dq +
dt
p
q
t

while the exact differential of the right-hand side (rhs) of (1.38) is given by
L
L
L
dq
dq
dt
q
q
t
L

(1.40)
dt.
= q dp pdq

t
Here we used (1.38). Because rhs of (1.39) equals to rhs of (1.40), we immediately
recover the Hamilton equations (1.1) from the common terms of dp and dq. Thus
we started from the Lagrange equations (1.37) together with (1.38) and derived
the canonical Hamilton equations (1.1). It is straightforward to show that (1.1)
with (1.1) results in the Lagrange equations (1.37). It proves the equivalence of the
Lagrangian and the Hamiltonian mechanics. Also the terms with dt results in the
following relation
d (pq L) = p dq + q dp

(1.41)

L
H
=
.
t
t

12

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

In most cases below both H and L do not explicitly depend on t meaning that
(1.41) is zero. Such systems are called autonomous.
For the motion of particles in the potential (1.2) we obtain from the Legendre
transform (1.38) expressing p through q that
(1.42)

L=

N
X
mi qi 2
i=1

U (q).

Notice that the potential energy U enters with the opposite sign in comparison with
the Hamiltonian (1.2).
1.7. Holonomic constraints and the Lagrangian mechanics on manifolds
The analysis of numerous mechanical systems can be greatly simplified if we
consider ideal objects like rigid body. The planar pendulum considered in Section
1.1 is the example of such simplification with the ideal rigid massless rod attached
to the point mass. That system can be considered as the limit of the Newton
equations of the potential motion where the mass of the rod approaches zero while
its stiffness (represented by the potentials for its stretching, bending and twisting)
goes to infinity. That limit can be taken rigorously (see e.g. [Arn89]). Such type of
constrains are called by holonomic constraints. These constraints can also depend
on time t.
Consider the general system of Np point masses in the D-dimensional real vector
space RD (usually D = 1, 2, 3) as in equation (1.2). The positions of these masses
are given by vectors r1 , r2 , . . . , rK RD , i.e. by a point in the real vector space
RNp D . Applying Nh independent holonomic constraints we restrict the allowed
values of these coordinates to Np D Nh -dimensional surface M in RNp D . From
point of view of equation (1.2) it means that we take the limit of the potential U
to be infinitely large if the holonomic constraints are not satisfied. Assume that
N = Np DNh and q := (q1 . . . , qN ) is the set of generalized coordinates on M . The
dynamics of Np point masses with Nh constraints is equivalent to the Lagrangian
mechanics for the generalized coordinates q on the surface M [Arn89].
A planar motion of a point mass (Np = 1 and D = 2) without constraint is
characterized by two coordinates (e.g. vertical and horizontal coordinates). The
planar pendulum considered in Section 1.1 has one holonomic constraint because
the rigid rod allows for the mass to move on the circle S 1 only. The position of the
point mass on that circle is fully characterized by a single generalized coordinate
q = .
The surface M is called the configuration space of the Lagrange equations
(1.31). M is the N -dimensional differential manifold. It means that the small
neighborhood of each point of M is homeomorphic to the N -dimensional Eucledian
space EN as well as that M has a globally defined differential structure (see Appendix ?? for the detailed definition). The Eucledian space EN is the vector space RN
PN
together with the standard scalar product x y = i=1 xi yi for any x, y RN . The
dimension N of the configuration space is called the number of degrees of freedom.
Below we often consider the dynamical systems with constraints which do have
any simple mechanical analogies and do not allow easy derivation of the constraints
from the unconstrained systems. However the notion of constraints remains extremely helpful for many Hamiltonian systems.

1.7. HOLONOMIC CONSTRAINTS AND THE LAGRANGIAN MECHANICS ON MANIFOLDS13

Consider a smooth curve q(t) on the N -dimensional configuration space M

parameterized by a scalar t R. Then q(t)

is the tangent vector to M . The
set of all tangent vectors at point q (can be obtained by parameterization of all
curves on M which pass through q) forms the N -dimensional vector space T Mq
which is called the tangent space to M at the point q. If M is embedded into
the Eucledian space ENe (as in the case of Np point masses described above with
Ne = Np D) then T Mq can be equivalently defined as the orthogonal complement to the set of vectors {f1 , f2 , . . . , fNe N }. Here Ne N scalar functions f1 (x), f2 (x), . . . , fNe N (x), x ENe define the configuration space M by
Ne N holonomic constraints f1 (x) = 0, f2 (x) = 0, . . . , fNe N (x) = 0. These
constraints are assumed to be independent at M meaning that the set of vectors
{f1 , f2 , . . . , fNe N } is linearly independent at each q M .
The union of all tangent spaces T Mq for all q M is called the tangent bundle
of M and is denoted by T M . Each point of T M consist of the point q M and the
tangent vector to M at q. T M is 2N-dimensional differential manifold. Assume
that (1 , 2 , . . . , N ) are the components of in the local coordinates (q1 , q2 , . . . , qN )
of M . Then (q1 , q2 , . . . , qN , 1 , 2 , . . . , N ) forms a local coordinate system in T M.
on the manifold can be conFor autonomous systems the Lagrangian L(q, q)
sidered as the mapping from T M into R. Similarly, for nonautonomous systems the
t) is the mapping T M R R. The least action principle and all
Lagrangian L(q, q,

equations of Section 1.31 are valid assuming that q(t) M and (q(t), q(t))
TM.
defining the Lagrangian mechanics on manifolds.
Assume that x ENe are the coordinates for the Newtonian equations of the
potential motion of point masses given by the equations (1.42) and (1.31). Consider
the coordinate transform
(1.43)

x = g(q)

from the generalized coordinates q RN to x, where N is the dimension of the

gi
to be N for
configuration space M . We choose the rank of the Jacobian matrix q
j
any q M . By the time derivative of (1.43) we obtain the relation between the
velocities q and x in both coordinate systems as follows
X
gi
xi =
q
(1.44)
.
q
j=1
Then the kinetic energy T =

Ne
P
i=1

mi xi 2
2

is transformed into the quadratic form for

q and the Lagrangian (1.42) takes the following form

(1.45)

L=

N
X
Aij (q)qi qj
U (q),
2
i,j=1

where
(1.46)

Aij =

Ne
X
k=1

mk

gk (q) gk (q)
qi
qj

are the elements of the symmetric N N matrix A = AT (AT means the transposed
matrix of A). It follows from (1.46) and positivity of masses mi > 0, i = 1, 2, . . . , Ne
that A positive-definite matrix (i.e. T A > 0 for any nonzero column vector
RN ). We also abused a notation by setting U (q) = U (x).

14

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

We use the example (1.45) to define the kinetic energy of in the Lagrangian
mechanics as the general symmetric positive-definite quadratic form in q:
N
X
Aij (q)qi qj
T =
.
2
i,j=1

(1.47)

The potential energy U is defined as the general function of q. The Lagrangian

equations have the natural form if L = T U. Then the Lagrange equations take
the following form
(1.48)

N
X

Aij (q)qj + other terms = 0,

i = 1, 2, , . . . , N,

j=1

where other terms stands for the terms which depend of q and q only but not
on higher derivatives of q over time. The symmetry the positive-definiteness of A
which provides the
ensures its invertibility. It means that (1.48) is solvable for q
existence and the uniqueness of the solutions of the Cauchy problem for the natural
Lagrange equations.
More general non-natural Lagrange systems have a Lagrangian L which is a
general general function of q and q (and t for nonautonomous systems). Then it
is generally impossible to separate the Lagrangian into the kinetic and potential
energy. The existence and the uniqueness of the solutions of the Cauchy problem
2
L
to be nonsingular on M .
for such systems require that the matrix qi q
j
1.8. Oscillations
Consider the equilibrium point q0 M of the Lagrangian system (1.31), i.e.
q=q0 = 0. For the natural Lagrangian system (1.45),(1.31) it implies that q0 is
q|
the stationary point of U (q),

U (q) 
(1.49)
= 0.
q 
q=q0

Consider the motion in the neighborhood of the equilibrium point q0 . Assume

that both q q0 and q are small and of the same order and expand the Lagrangian
(1.45) in the Taylor serious near q = q0 and q = 0. Without loss of generality we
set q0 = 0 (can be done by the translation of the generalized coordinates q) and
U (0) = 0 (addition of the arbitrary time-independent real number to U (q) does
not change the Lagrangian equations (1.31)). A first nontrivial contribution to the
Lagrangian (1.45) is quadratic in q and q as follows
(1.50)

L2 =

N
N
X
X
Aij qi qj
Bij qi qj

,
2
2
i,j=1
i,j=1

where L = L2 + O(q 2 q) + O(q3 ),

(1.51)

Aij := Aij (0)

and
(1.52)


2 U (q) 
Bij :=
qi qj q=0

1.8. OSCILLATIONS

15

are the constant symmetric N N matrices A and B, respectively. We also used

(1.49) to remove linear in q terms.
The Lagrange equations (1.31) for the Lagrangian L2 results in the system of
N linear ODEs of the second order
A
q = Bq.

(1.53)

The kinetic energy T = 12 q Aq and the potential energy U = 12 q Bq in the

Lagrangian L2 are the quadratic forms in independent variables q and q. The
kinetic energy is the positive-definite quadratic form because A is the positivedefinite matrix, as we found in Section 1.7.
Consider the quadratic forms q Aq and q Bq. According to the standard
theorem of linear algebra (see e.g. [?]), these two quadratic forms, one of which is
positive-definite, can be simultaneously converted to the sum of squares by a linear
nonsingular transformation
(1.54)

Q = Cq,

where C is the N N matrix of the transformation and Q RN . For the quadratic

form q Aq we can use the same transformation
= Cq,

(1.55)

as follows
which results in the expression for L2 in new variables Q and Q
N

(1.56)

L2 =


1 X 2
Qi i Q2i
2 i=1

and the Lagrangian equations (1.53) turn into the the following decoupled system
of ODEs:
Qi = i Qi .

(1.57)

Here 1 , 2 , . . . N are the eigenvalues of B with respect to A determined by the

characteristic equation
(1.58)

det(B A) = 0.

Notice that the positive-definiteness of A ensures its the invertibility and allows
to rewrite (1.58) as the usual characteristic equation det(A1 B I) = 0 for the
matrix A1 B, where I is the identity matrix.
The quadratic Lagrangian (1.56) is in the form (1.42) which implies that the
Hamiltonian of that system is
N

(1.59)

H2 =

1X
Pi 2 + i Q2i ,
2 i=1

where the canonical momenta are

(1.60)

Pi = Q i , i = 1, . . . , N.

16

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

Exercise 1.
Show that the transformation from q and p to Q and P defined by (1.50),
(1.55), (1.56) and (1.60) is the canonical transformation for the Hamiltonian equations (1.1) with the quadratic Hamiltonian (1.56).
The Hamiltonian (1.4) for the harmonic oscillator in physical variables is transformed into (1.71) by the following trivial change of variables
Qj = m1/2 qj , Pj = m1/2 pj , j = 1, . . . , N,

(1.61)

which results in the Hamiltonian (1.59) with j = j2 , j = 1, . . . , N .

For positive eigenvalues j = j2 > 0 we recover from (1.57) a set of decoupled harmonic oscillators as in the equation (1.3) with the general solution
Qj = aj cos (j t + j ), similar to (1.5). Zero eigenvalues j = 0 correspond to
the general solution Qj = cj,1 t + cj,2 , where c1 and c2 are the arbitrary real constants. The general solution for the negative eigenvalues j = j2 < 0 is given
by Qj = cj,1 ej t + c2,j ej t . Using the transformation (1.55) we write the general
solution of the Lagrange equations (1.53) as follows
q=

N1
X

aj cos (j t + j )qj +

j=1

NX
1 +N2

cj,1 ej t + c2,j ej t qj

j=N1 +1

(1.62)

N
X

(cj,1 t + cj,2 ) qj ,

j=N1 +N2 +1

where qj is jth eigenvector of A1 B, N1 is the number of positive eigenvalues,

N2 is the number of negative eigenvalues and N N1 N2 is the number of zero
eigenvalues for A1 B with all eigenvalues counted according to their algebraic
multiplicity. Each term in (1.62) is the particular solution of (1.53) called by
a normal mode of (1.53). Notice that although some of j can have algebraic
multiplicity more than one, their algebraic multiplicity is always equal to their
geometric multiplicity, i.e. eigenspace of all eigenvalues spans RN . It immediately
follows from the simultaneous transformation by (1.55) of both quadratic forms
q Aq and q Bq to the sum of squares. It also implies that no extra power of
t appears in (1.62) for Lagrange equations (contrary to the case of general linear
ODE system with repeated eigenvalues).
If q = 0 is the strict minimum of the potential energy then the matrix B is
positive-definite. It implies that all eigenvalues j are positive and the equation
(1.62) reduces to the sum of oscillating terms only
(1.63)

q=

N1
X

aj cos (j t + j )qj .

j=1

Any solution of the Lagrange equations is a superposition of oscillations in that

case.
Example. Consider the Newton dynamics of a molecule which consists of Np
atoms with different masses mk > 0 located at points r1 , r2 , . . . , rNp R3 . All
atoms can move with respect to each other (i.e. there are no holonomic constraints).
The configuration space is the N = 3Np -dimensional Eucledian space EN . Assume
that r1,0 , r2,0 , . . . , rN/3,0 are equilibrium positions of the atoms in the molecule.
Choose a vector q = (r1 r1,0 , r2 r2,0 , . . . , rN/3 rN/3,0 ) EN as the vector of the

1.9. HARMONIC OSCILLATOR AND COMPLEX VARIABLES

17

configuration space so that q = 0 is the equilibrium. Consider small deviations from

that equilibrium which allows to use the quadratic Lagrangian (1.50). The absence
of the holonomic constraints implies that the kinetic energy has the diagonal form
N/3
N
P mk
P
m
j qj 2
T =
(q
+q
+q
)
=
3k2 = m
3k1 = m
3k := mk is
3k2
3k1
3k
2
2 . Here m
j=1

k=1

the mass the kth atom. The potential energy has the general form U =

N
P
i,j=1

Bij qi qj
2

as in (1.50) and the Lagrangian is given by

q Mq
q Bq

,
2
2
where M = A is the diagonal matrix with the main diagonal (m
1, . . . , m
N ). The
Hamiltonian is given by
L=

(1.64)

(1.65)

H=

N
N
X
1 X
pj 2
+
Bij qi qj ,
2m
j
2 i,j=1
j=1

where the momentum p = Mq.

The Hamilton equations for (1.65) (or equivalently the Lagrange equations for
(1.64)) result in the linear system
M
q = Bq.

(1.66)

Assume additionally that all atoms interacting only pairwise with each other by
the harmonic potentials with the positive interaction constants ij = ji > 0. The
Hamiltonian of that system takes the following form
H=

(1.67)
N
P

In this case Bii =

N
N
X
pj 2
1 X
+
ij (qi qj )2 .
2
m

4
j
j=1
i,j=1

ij and Bij = ij , for i 6= j with i, j = 1 . . . , N . It also

j=1, j6=i

immediate follows from (1.67) that the matrix B is non-negative (i.e. x Bx 0 for
any x RN ) and respectively all j are nonnegative. Zero values of j have a simple
physical meaning representing the translation of the molecule in any direction.
There are three independent directions for such translation (i.e. for the motion of
the molecules center of mass). The general solution (1.62) is then reduced to
(1.68)

q=

N
3
X

aj cos (j t + j )qj +

N
X

(cj,1 t + cj,2 ) qj .

j=N 2

j=1

1.9. Harmonic oscillator and complex variables

Consider the quadratic Hamiltonian (1.59) and assume that all i are positive,
i = i2 , i.e. all normal modes are oscillations. It is often convenient to introduce in
(1.59) complex variables aj and their complex conjugate a
j , j = 1, . . . , N as follows
1
(j Qj + iPj ) ,
(2j )1/2
1
a
j =
(j Qj iPj ) ,
(2j )1/2
aj =
(1.69)

18

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

where we choose the positive sign j > 0 for all j. Solving (1.69) for Qj and Pj we
obtain that
1
Qj =
(aj + a
j ) ,
(2j )1/2
(1.70)
 1/2
j
Pj = i
(aj a
j ) .
2
The Hamiltonian (1.59) simplifies by (1.69) into
H2 =

(1.71)

N
X

j |aj |2 ,

j=1

where the positive value of H2 is ensured by our choice j > 0.

In a similar way, the Hamiltonian (1.4) of the harmonic oscillator is transformed
into (1.71) by the following change of variables
1
(mj j qj + ipj ) ,
aj =
(2mj j )1/2
(1.72)
1
(mj j qj ipj ) .
a
j =
(2mj j )1/2
Consider now complex variables (1.69) for a general time-independent Hamiltonian H(P, Q), where 1 , . . . , N are N arbitrary positive constants. Using the
Hamiltonian equations (1.1) together with (1.1) we obtain that




1
dPj
1
daj
dQj
H
H
=
+i
=
j
j
i
,
dt
dt
dt
Pj
Qj
(2j )1/2
(2j )1/2




(1.73)
1
dPj
d
aj
dQj
1
H
H
=
j
i
j
=
+i
.
1/2
1/2
dt
dt
dt
Pj
Qj
(2j )
(2j )
Expressing Pj and Qj in H through aj and a
j by (1.70) we obtain the complex
form of the Hamiltonian equation for the pairs of independent variables aj and a
j
as follows
H
daj
= i
.
(1.74)
dt

aj
The equation for
real function:
(1.75)

daj
dt

is obtained the complex conjugation if we recall that H is the

d
aj
H
=i
.
dt
aj

Calculating partial derivatives in (1.74) and (1.75) we imply a j0 = 0 and ajj = 0

j
for any j, j 0 = 1, . . . , N because we consider aj and a
j as independent variables.
For the particular case of the quadratic Hamiltonian (1.71) we obtain from
(1.74) the system of first order complex ODEs
daj
= ij aj , j = 1, . . . , N,
dt
which has a solution aj (t) = aj (0)eij t representing the complex form of the
solution of N decoupled harmonic oscillators.
Complex variables aj and a
j are the classical analogs of the Bose creation and
annihilation operators of quantum mechanics (see e.g. [LL76]). The transformation
(1.76)

1.10. COMPLEX VARIABLES IN WEAKLY NONLINEAR CASE

19

(1.69) is the classical analog of the quantum mechanical transformation from the
coordinate-momentum representation to the representation by the Bose creation
and annihilation operators.
1.10. Complex variables in weakly nonlinear case
Complex variables (1.70) are especially useful if the Hamiltonian can be expanded in a power series of the canonical variables Pj and Qj . E.g. it occurs if we
expand the Hamiltonian in the power series near the stationary point (1.49) and
take into account next terms beyond quadratic terms considered in Section 1.8.
The linearity of the transformation (1.70) implies a power series in variables aj and
a
j for the same Hamiltonian expanded in the power series.
Consider a nonlinear oscillator with the Hamiltonian
p2
(1.77)
H=
+ U (q),
2m
such that the potential U (q) has a minimum at q = 0. Without loss of generality
we set U (0) = 0 because the Hamiltonian equations (1.1) do not change if shift
U (q) by an arbitrary real constant.
Assume that q is small and expand U (x) in a Taylor series about q = 0:
m 2 2
q + q 3 + q 4 + . . . ,
2
where we define in the same way as in (1.3) so it would have a meaning of
frequency of small oscillations.
The Hamiltonian has now a form of power series in canonical variables p and q
as follows
p2
m 2 2
H2 =
+
q ,
2m
2
3
H3 = q ,
(1.79)
(1.78)

U (q) =

H4 = q 4 ,
...,
where subscripts in H represent the order of terms in powers of q an p.
Limiting to H = H2 we recover the harmonic oscillator considered in Section
1.9 with the dynamical equation (1.9). Substituting (1.70) into (1.79) we obtain
that
(1.80)


H3 =

1
2m

3/2

(a3 + a
3 + 3a
a2 + 3
aa2 )




U 3
a +a
3 + V |a|2 a
+ |a|2 a ,
3

which results by (1.74) in the following Hamiltonian equation for H = H2 + H3 :

(1.81)

da
+ ia = iU a
2 iV a2 2iV a
a,
dt

where

(1.82)

U = V = 3

1
2m

3/2
.

20

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

Similar, at the next order H = H2 + H3 + H4 we obtain from (1.70) and (1.79) that

2
1
4
H4 = q =
(1.83)
[a4 + a
4 + 6|a|4 + 4|a|2 (
a2 + a2 )]
2m
and the corresponding dynamic equation takes the following form
(1.84)

2
da
1
+ ia = iU a
2 iV a2 2iV a
a i
(4
a3 + 12|a|2 a + 12|a|2 a
+ 4a3 ).
dt
2m
1.11. Nonlinear oscillator and the generation of multiple harmonics
To analyze the equation (1.84) we note that its leading order solution in the
limit of small a is reduced to the harmonic oscillator equation da
dt + ia = 0 with
the exact solution a = C1 eit . We call that solution by a first (or fundamental)
harmonic. We assume that nonlinear terms in r.h.s of (1.84) result in slow time
dependence of C1 at times  1/ which motivates looking for the solution of (1.84)
in the following form
(1.85)

a = C1 eit + C0 + C2 e2t + C2 e2it + . . . ,

where C1 (t), C0 (t), C2 (t), C2 (t) are the slow functions of time. We refer to
C0 (t) and C2 (t) as the amplitudes of the zeroth and second harmonics, respectively.
C2 (t) can by called by the minus second harmonic but instead we also refer it as
the second harmonic keeping in mind that if we have a harmonic with a frequency
j then nonlinearity immediately results in the formation of j harmonic. All
these harmonics are generated by the quadratic nonlinear terms in r.h.s of (1.84).
It is seen if we substitute a in r.h.s. (1.84) by the leading order harmonic solution
a = C1 (t)eit and neglect for now cubic terms which gives
(1.86)

da
+ ia = iU C12 e2it iV C1 2 e2it 2iV |C1 |2
dt

In other words, our linear oscillator (represented by l.h.s of (1.86) is forced by

the terms with frequencies 2, 2 and 0 at r.h.s of (1.86). We now replace a
in l.h.s of (1.86) by its expansion (1.85) in harmonics and collect all terms with
frequencies 0, 2. At these frequencies at leading order we can neglect the time
derivatives of C0 (t), C2 (t), C2 (t) in (1.86) which results in the close expressions
for these amplitudes as follows
(1.87)

C0 =

2V
V
U
|C1 |2 ; C2 = C12 ; C2 = C 2 .

These second and zero harmonics are of second order, |C1 |2 in amplitude of the
fundamental harmonic. Retaining slow time dependence of C1 (t), C0 (t), C2 (t), C2 (t)
2
results in the addition of small terms d|Cdt1 | 1  |C1 |2 into r.h.s, of each expression in (1.87). In a similar way, extending (1.85) to include third, C3 , fourth,
C4 etc harmonics and substitution into (1.86) allows to conclude that at leading
order |C3 | |C1 |3 , |C4 | |C1 |4 etc. Thus the nonlinearity in (1.84) results in the
generation of the harmonics (1.84) at multiple frequencies 0, 2, 3, 4, . . ..
These higher harmonics also modify the dynamics of the first harmonic C1 . To
understand that modification we return to the equation (1.81), collect all terms

1.11. NONLINEAR OSCILLATOR AND THE GENERATION OF MULTIPLE HARMONICS 21

which eit corresponding to the fundamental harmonic and use (1.87) to express C0 (t), C2 (t), C2 (t) through C1 (t). That procedure can be qualitatively
interpreted as the projection of a into the state eit and gives
dC1
= 2iU C2 C1 i2V C1 C0 2iV C1 C0 2iV C2 C1
dt
20
(1.88)
= i V 2 |C1 |2 C1 ,
3
where we used that U = V .
Returning back from C1 to the full form of the fundamental harmonic a1
C1 (t)eit results in the equation
(1.89)

da1
+ i( +
)a1 = 0
dt

where
20 V 2
|a1 |2 .
3
The expressions (1.89) and (1.90) are however not sufficient because we also
need to find a similar contribution to from H4 . For that we take into account the
term with in (1.84) and again project that term on the state eit which results

2
1
12|C1 |2 C1 in
in the equation similar to (1.88) but with the added term i 2m
r.h.s. Thus instead of equations (1.89) and (1.90) we obtain a full expression
(1.90)

da1
+ i( + )a1 = 0
dt
which includes all terms of the order O(|a1 |2 a1 ). Here is called by the nonlinear
frequency shift and is given by
" 
#
2
1
20 V 2
(1.92)
= 3

|a1 |2 .
m
3

(1.91)

The equations (1.91) and (1.92) imply that R and

solve (1.91) and (1.92) explicitly and obtain that
(1.93)

d|a1 |2
dt

= 0. It allows to

a1 (t) = a1 (0)ei(+)t ,

which has the same form as the solution a1 (t) = a1 (0)eit for the harmonic oscillator (1.76) except that the frequency is replaced by + .
Equation (1.91) has the complex Hamiltonian form (1.74) with the Hamiltonian
T
|a1 |4 := H2 + H4,ef f ,
2
where H2 = |a1 |2 is the quadratic part of the Hamiltonian and

2
1
20 V 2
(1.95)
T := 3

m
3
(1.94)

H = |a1 |2 +

is the interaction constant for the effective 4th-order Hamiltonian H4,ef f . There


1 2
are two contributions to T . The first one is given by 3 m
and can be immediately obtained from the averaging of the original Hamiltonian H4 : hH4 i =


3
1 2
|a1 |4 , where h. . .i means averaging over fast oscillations at time scale 1/.
2 m
V2
The second contribution, 10
3 , comes from taking into account of the zero and

22

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

second harmonics in the cubic terms H3 (1.80) as explained above. That contribution corresponds to the second order of the perturbation theory (at the first order
we calculated C0 , C2 , C2 in (1.87) and at the second order we found the influence
of these terms on the dynamics of a1 ). In other words, the interaction constant


1 2
3 m
of hH4 i is renormalized by the cubic terms H3 of the Hamiltonian in the
second order of the perturbation theory. That renormalization has the negative
sign which is consistent with the well-known fact of the quantum mechanics that
the shift of the ground state level from the second order perturbation theory is
negative one [LL76].
We conclude that there are two effects of nonlinearity of the oscillator:
1. Nonlinearity results in the formation of multiple harmonics.
2. Nonlinearity produces the nonlinear frequency of the fundamental harmonic.
The above technique of obtaining (1.93) represents the averaging method (averaging
of over fast oscillations eit assuming weak nonlinearity results in the calculation
of the nonlinear frequency shift (1.92)).
Exercise 2.
Find the nonlinear frequency shift for the pendulum (1.6).
1.11.0.1. Resonance in nonlinear oscillator. Consider a dynamics/exitation of
the nonlinear oscillator with an external periodic forcing. In the linear approximation the equation (1.91) with the added forcing if0 eit takes the following
form
a1
+ i( i)a1 = if0 eit ,
t
where f0 C and R are constants (i.e. the forcing is monochromatic). Also
we added a small damping coefficient > 0 (  ) into l.h.s. of equation (1.96).
A solution of (1.96) is the sum of the transient solution eitt (corresponds to the solution of the homogeneous part with forcing excluded of (1.96))
and the particular solution of full nonhomogeneous equation of (1.96). Assuming
t , the transient solution can be neglected and the remaining solution takes
the following form

(1.96)

f0
eit ,
+ i
which gives a time-independent resonance solution in the form of Lorentzian function for |a1 |2 as follows
a1 =

|a1 |2 =

(1.97)

|f0 |2
.
( )2 + 2

It follows from here that |a1 |2 at 0 and .

Taking into account nonlinearity, we replace (1.96) by
a1
+ i( i)a1 = iT |a1 |2 a1 if0 eit ,
t
which modifies (1.97) into

(1.98)

(1.99)

|a1 |2 =

|f0 |2
,
( T |a1 |2 )2 + 2

:= ,

EXERCISE 4.

23

where is the frequency detuning between the linear oscillator frequency and the
pumping frequency .
Equation (1.99) represents the cubic equation for the unknown amplitude |a1 |2
which remains bounded from all values of , and f0 . To see that we assume by
contradiction that |a1 |2 for any fixed , and f0 which is inconsistent with
(1.99) (l.h.s while r.h.s would 0 in that case). Thus the nonlinear frequency
shift regularizes the linear resonance. At that resonance 0 and 0 which
implies from (1.99) that
2/3

|f0 |
2
2
.
|a1 | |a1 |sat :=
T
Also |a1 |2 is close to |a1 |2sat for fixed > 0 and 0 provided T |a1 |2 )  .
Taking the limit f0 0 for each fixed values of > 0 and we recover the
linear solution (1.97). In that limit there is only one real solution of (1.99) for
|a1 |2 . If we increase |f0 | then for large enough amplitude |f0 | > |f0cr |, there are
three real solutions of (1.99) for |a1 |2 , where |f0 | = |f0cr | is the critical value of the
forcing |f0 | corresponding to the appearance of two additional real roots. +
Exercise 3.
Find |f0cr |.
Answer: |f0cr |2 =

2 3
T .

Exercise 4.
Determine the stability (in time) of all three roots |f0 | > |f0cr |. Find |f0cr |.
1.11.0.2. Parametric resonance. Another type or resonance occurs if the oscillator frequency has a periodic time dependence. Simplest example of the periodic
time dependence is given by
2 (t) = 02 (1 + cos t),
where  1 is the small parameter. The dynamics of oscillator without damping
is given by the Mathieus differential equation
d2 q
= 2 (t)q,
dt2
where q is the coordinate. To bring that equation to the complex form we introduce
the momentum as p = q (for simplicity we set the mass m = 1) an define a and a

similar to (1.72) as
(1.100)

1
(0 q + ip) ,
(20 )1/2
1
a
=
(0 q ip) ,
(20 )1/2
a=

(1.101)

except that now time-independent part 0 of the frequency (t) is used for that
transformation.
Consider the following Hamiltonian
0
(1.102)
H = 0 |a|2 +
(a + a
)2 cos(t),
4

24

1. FINITE DIMENSIONAL CANONICAL HAMILTONIAN SYSTEMS

then the Hamiltonian equations (1.74) and (1.75) result in

da
0
= i0 a i
(a + a
) cos(t),
dt
2
(1.103)
d
a
0
= i0 a
+i
(a + a
) cos(t).
dt
2
Solving that systems of ODEs for q results in equation (1.101). Thus the Mathieus
differential equation(1.101) is equivalent to the Hamiltonian equations (1.74) and
(1.102).
We now solve (1.74) and (1.102) assuming that the leading order solution is
given by
i0 t

a(t) = C(t)ei0 t and a

(t) = C(t)e
,
where C(t) is the slow function of time in comparison with ei0 t . We separate the
Hamiltonian (1.102) into the harmonic part H2 = 0 |a|2 and the interaction part
0
(a + a
)2 cos(t).
Hint =
4
We again use averaging of Hint over the fast time scale 1/0 . Only nonoscillating
terms survive the averaging. Recalling that cos t = (eit +eit )/2 as well as that
at leading order a ei0 t and a
ei0 t , we conclude that many terms in Hint
oscillate and average out to zero, such as |a|2 cos t. Nonzero average is possible for
a2 eit provided 20 ' which is near to the condition
(1.104)

20 =

of the parametric resonance. In that case we obtain that


0  2 it
a e +a
2 eit .
(1.105)
hHint i =
8
Also hH2 i = H2 = 0 |a|2 , i.e. the harmonic part of the Hamiltonian is not affected
by the averaging.
The parametric resonance condition (1.104) has a simply analog

in quantum
2 + h
mechanics. We rewrite equation (1.105) as hHint i = 20 ha
a2 , where h(t) :=
 it
is the pumping amplitude. Recall that a and a
are the classical analogs of
4e
the annihilation and creation operators, respectively. Then the term h
a2 in hHint i
describes the annihilation of a quantum of pumping and the creation of two quanta
of the oscillator. That process requires the energy conservation which is insured by
the condition (1.104).
Consider the dynamical equations for the averaged Hamiltonian hHi = H2 +
hHint i (1.103) and (1.105):
da
hHi
0 it
+ a = i
= i0 a i
a
e
,
dt

a
4
(1.106)
d
a
hHi
0 it
+
a=i
= i0 a
+i
ae ,
dt
a
4
where we added a linear damping term a into l.h.s., similar to Section (1.11.0.1).
Comparison between first equations in (1.103) and (1.106) shows that (1.106) does
not include terms which oscillate with a different frequency than ei0 t . Taking
into account such non-resonant terms in (1.103) can be done by the expansion
in multiple harmonics, qualitatively similar to the expansion (1.85). These terms
however would modify (1.106) by the inclusion of O(2 ) terms which we neglect here.

1.14. SYMPLECTIC LEAVES. EXAMPLE OF FREE MOTION OF RIGID BODY

25

Thus the averaging of the Hamiltonian is equivalent to the neglect of non-resonant

terms in the dynamic equations (1.103).
We introduce new variables c and c as follows
a = c(t)ei

t
2

, a
= c(t)e+i

t
2

which transform (1.106) into ODE system with constant coefficients

 


c

0
+ i 0
+ c+i
c = 0,
t
2
4
 


(1.107)

0
+ i 0
+ c i
c = 0.
t
2
4
Assuming c, c et we obtain from (1.107) a homogeneous system of linear equations



0

c = i
c,
+ + i 0
2
4



(1.108)

0
+ i 0
c = i
c
2
4
for the unknowns c and c. That system is solvable provided the matrix of its
coefficients has a zero determinant which gives
s

2
(0 )2

0
.
(1.109)
=
16
2
It follows from (1.109) that the instability ( > 0) is possible for  > cr =
4/0 provided 0 is close enough to /2 to ensure that the expression under the
square root in (1.109) is positive as well as to overcome dissipation rate . This
instability is called the parametric instability. The growth rate of the parametric
instability reaches maximum if the condition (1.104) is exactly satisfied.
A difference between the forced oscillator of Section (1.11.0.1) and the parametric amplification of Section (1.11.0.2) can be seen from the everyday experience
in playing on a childrens swing. Rocking back and forth pumps the swing as a
forced harmonic oscillator. Once moving, the swing can also be parametrically amplified by alternately standing and squatting at key points in the swing arc, i.e. by
periodically changing the moment of inertia of the swing and hence the resonance
frequency. The rate of that change is twice of the natural frequency of the swing
satisfying the parametric resonance condition (1.104).
Exercise 5.
Find a frequency width of the parametric resonance , i.e. the range of values
of at which a parametric excitation occurs assuming  > cr .
1.12. Canonical transformations
1.13. Generalization of the canonical Hamilton equations. Symplectic
structure. Poisson mechanics.
1.14. Symplectic leaves. Example of free motion of rigid body

Bibliography
[Arn89] V. I. Arnold, Mathematical methods of classical mechanics, Springer, 1989.
[LL76] L. D. Landau and E. M. Lifshitz, Quantum mechanics (course of theoretical physics,
volume 3), Butterworth-Heinemann, New York, 1976.
[LL89]
, Mechanics (course of theoretical physics, volume 1), Pergamon Press, New York,
1989.

27

Index

averaging, 22
frequency detuning, 23
interaction constant, 21
kinetic energy, 4
nonlinear frequency shift, 21
parametric instability, 25
potential energy, 4
smooth manifold, 4

29