Applications of Canonical Transformations in Hamiltonian Mechanics Brian Tu Jan. 14, 2014
Contents 1 Intro duction
2
2 Preliminaries
2
3 Poisson Bracket
3
3.1 3.2
Characterization of Canonical Transforms . . . . . . . . . . . . . . . . . Application to Integrals of Motion . . . . . . . . . . . . . . . . . . . . . .
3 5
4 Infinitesimal Canonical Transformations
5
5 Generating Functions
6
1
1
Introduction
Canonical transformations were introduced in the theory of Hamiltonians as a class of transformations that preserve the form of the Hamiltonian equations. We explore these transformations in greater detail, looking at topics such as its invariants, generating such transformations, and time evolution of a Hamiltonian system as a canonical transformation. We consider only time-independent Hamiltonians.
2
Preliminaries
We begin with a few definitions for review. Definition 2.1. Recall that the Hamiltonian H is a function of 2n coordinates q = (q 1 , q 2 , . . . , qn ) and p = ( p1 , p2, . . . , pn) that satisfies the 2n relations
∂H ∂q i ∂H q ˙i = , ∂p i
p˙i = −
where 1 ≤ i ≤ n, known as Hamilton’s equations. Definition 2.2. A transformation of coordinates is canonical if it preserves the form of Hamiltonians equations [3, p. 14]. Formally, let x = (q 1 , . . . , qn , p1 , . . . , pn ). Let
J =
0 1 −1 0
be a 2n × 2n block matrix, e.g., 0 is an n × n 0 matrix, 1 is I n, and -1 is −I n . Hamilton’s equations then can be written as x˙ = J ∇H, Let y : x i → yi (x) be a transformation of coordinates. We have 2n
2n
∂H ∂y ∂y y˙ = x˙ = J , i
i
j =1
∂x j
i
j
j =1
∂x j
∂ x
j
and hence, y˙ = (J J J T )∇H,
where (J )ij = ∂yi /∂x j is the Jacobian of y. The form of Hamilton’s equation is preserved if J J J T = J. (1) Hence a transformation is canonical if its Jacobian satisfies (1). Incidentally, such a matrix J is called symplectic.
2
3 3.1
Poisson Bracket Characterization of Canonical Transforms
We saw above that one way to characterize a canonical transformation is by checking (1). Another way of characterizing these transformations is by using an operation called the Poisson bracket . Definition 3.1. A Poisson bracket is a binary operation on two functions f (q, p),
g(q, p) defined as
n
∂f ∂g ∂f ∂g {f, g } = − . ∂q i ∂p i
i=1
∂p i ∂q i
It is immediate from the definition that
{f, g } = −{g, f } and that q and p satisfy
{q i , q j } = 0,
{ pi , p j } = 0,
{q i , p j } = δ ij .
(2)
It is natural to expect to expect that (2) holds for canonical transformations; in fact this is a sufficient condition. We have the following theorem. Theorem 3.1. The Poisson bracket is invariant under canonical transformations. Con-
versely, if Q : q i → Qi (q , p),
P : p i → P i (q , p)
is a transformation of q and p such that
{Qi , Q j } = 0,
{P i , P j } = 0,
{Qi , P j } = δ ij ,
then the transformation is canonical [4, p. 22]. Proof. For the first part, let f (q , p) and g(q , p) be any two functions. We have n
∂f ∂g ∂f ∂g {f, g } = − i=1
=
∂q i ∂p i
∂p i ∂q i
∂f ∂g J . ∂ x ∂ x
If we have a transformation y(x) with Jacobian J , then we have ∂f ∂f = J ∂ x ∂ y and thus ∂f ∂g J J J T ∂ x ∂ y ∂f ∂g = J . ∂ x ∂ y
{f, g } =
3
This shows the invariance of the Poisson bracket. For the other direction, note that we can write the Jacobian as ∂ Q ∂ Q ∂ q ∂ p . J = ∂ P ∂ P ∂ q ∂ p Computing, we have {Q, Q} {Q, P } J J J T = {P, Q} {P, P }
where ({Q, P })ij = {Qi , P j }, and the other entries are defined similarly. Thus we have
J J J T = J and the transformation is canonical. We develop another method of characterizing canonical transformations. Let Q (q , p) and P (q , p) be a canonical transformation. We have
and
∂Q i ∂Q i ∂Q i ∂H ∂Q i ∂H Q˙ i = q˙ + p˙ = − ∂ q ∂ p ∂ q ∂ p ∂ p ∂ q
(3)
∂H ∂H ∂ q ∂H ∂ p = + . ∂P i ∂ q ∂P i ∂ p ∂P i
(4)
Since the transformation is canonical, (3) equals (4), so we have ∂Q i ∂H ∂Q i ∂H ∂H ∂ q ∂H ∂ p ∂H = + − ⇒ ∂ q ∂ p ∂ p ∂ q ∂ q ∂P i ∂ p ∂P i ∂ p
∂Q
∂ p − ∂ q ∂P i i
∂H = ∂ q
∂
∂Q i + ∂P i ∂ p q
.
Since ∂H/∂ p = q = 0 and ∂H/∂ q = − p = 0, this implies that the quantities in the parentheses are 0. Hence ∂Q i ∂ p = ∂ q ∂P i
and
∂Q i ∂ q =− . ∂ p ∂P i
(5)
By equating P ˙i with ∂H/∂Qi , using a similar argument it can be shown that ∂P i ∂ q = ∂ p ∂Q i
and
∂P i ∂ p =− . ∂ q ∂Q i
(6)
Hence canonical transformations imply (5) and (6). In fact the converse is true as well. Suppose that (5) and (6) hold for some transformation Q and P not necessarily canonical. We have ∂H ∂H ∂ q ∂H ∂ p ∂H ∂Q i ∂H ∂Q i = + =− + = Q˙ i ∂P i ∂ q ∂P i ∂ p ∂P i ∂ q ∂ p ∂ p ∂ q and ∂H ∂H ∂ q ∂H ∂ p ∂H ∂P i ∂H ∂P i = + = + = P ˙i , ∂Q i ∂ q ∂Q i ∂ p ∂Q i ∂ q ∂ p ∂ p ∂ q and hence the transformation is canonical. Thus we obtain another characterization of canonical transformations [2, p. 9]. 4
3.2
Application to Integrals of Motion
The Poisson bracket also neatly characterizes the integrals of motion. A function f is an integral of motion iff {f, H } = 0 [4]. To see this, note that n
n
∂f ∂H ∂f ∂H ∂f ∂f df {f, H } = − = q ˙ + p˙ = , i=1
∂q i ∂p i
∂p i ∂q i
i=1
∂q i
i
∂p i
i
dt
which is 0 if f is an integral of motion. We can apply this to an integral of motion we discussed this year: Example 3.1 (Conservation of Linear Momentum). Recall that a coordinate q k is cyclic
if it does enter into the Hamiltonian; e.g., ∂H = 0. ∂q k Suppose that q k is a cyclic coordinate. Then momentum in that direction is conserved:
{ pk , H } = −
∂H = 0. ∂q k
Example 3.2 (Conservation of Energy). We know that that Hamiltonian is the sum of
the kinetic and potential energies of a system. We have {H, H } = 0 from antisymmetry of the Poisson bracket, and hence energy is conserved.
4
Infinitesimal Canonical Transformations
Here we consider canonical transformations that are parameterized by a single parameter. We willl see that the time evolution of a Hamiltonian system is a canonical transformation. Suppose that we have a family of canonical transformations indexed by a parameter λ, and suppose that the dependence on λ is continuous (think of them as, say, translations or rotations by a parameter). Furthermore, we can take λ such that at λ = 0 the transformation is the identity. That is, we have Qi (λ) = Q i (q , p, λ),
P i (λ) = P i (q , p, λ),
and P (0) = p .
Q(0) = q
If λ is very small, we can write Qi (λ) = q i + λdq i ,
P i (λ) = p i + λdpi .
We wish to know what relationships d q and d p must satisfy for the transformation to be canonical. The Jacobian is
J =
I + λ∂d /∂ n
q
q
λ∂d p/∂ q
5
λ∂d q /∂ p I n + λ∂d p/∂ p
which must satisfy J J J T = J . This gives us ∂d q ∂d p =− . ∂ q ∂ p If we could find a function F such that ∂F = dq and ∂ p
∂F = −d p, ∂ q
(7)
(8)
then (7) would be satisfied [5]. Note that the equations in (8) look exactly like Hamilton’s equations. Indeed, if we let λ be the time t, the transformation induced by the Hamiltonian H matches the differential equations for the flow of the system. Hence the advance of time in a Hamiltonian system is a canonical transformation. Now that we have established that time evolution of a Hamiltonian system is a canonical transformation, we can give a very quick proof of Liouville’s Theorem. Theorem 4.1 (Liouville’s Theorem). The phase flow of a Hamiltonian system is volume
preserving. Proof. Let t be fixed. Let the evolution of the Hamiltonian system after time t be a canonical transformation taking q to Q(q , p) and p to P (q , p). It suffices to show that
the Jacobian of the transformation is 1. Using properties of the Jacobian, we have ∂ (Q, P ) ∂ (q , p) ∂ (Q, P ) ∂ (q , p) = ∂ (q , P ) ∂ (q , P ) ∂ Q ∂ p = . ∂ q ∂ P
J =
and thus
∂ Q ∂ p = , ∂ q ∂ P
so we have
J = 1, as desired.
5
Generating Functions
We have seen various characterizations of canonical transformations, and a few examples of them. In fact, in the previous section on infinitesimal transforms we saw an example of a generating function. Here, we explore the construction of canonical transformations. Let F (q , Q) be any function between coordinates. If we define pi = 6
∂F , ∂q i
how can we define P i so that the resulting transformation is canonical? If we define P i = −
∂F ∂Q i
then F will be a canonical transformation. This can be shown using Theorem 3.1. F is known as a generating function of the first kind . There are four other kinds of generating functions that generate canonical transformations, with other coordinates defined as follows [5, 4]: ∂F 2 ∂F 2 , P = ∂ q ∂ P ∂F 3 ∂F 3 q = − , P = − ∂ p ∂ Q ∂F 4 ∂F 4 q = − , Q = . ∂ p ∂ P
F = F 2 (q , P ) − QP ,
p =
F = F 3 ( p, Q) + qp , F = F 4 ( p, P ) + qp − QP ,
I pledge my honor that this represents my own work in accordance with University policy. 7
References [1] Arnold, V. I. ”Variational Principles.” Mathematical Methods of Classical Mechanics . New York: Springer-Verlag, 1978. 66-70. Print. [2] Morii, Masahiro. ”Lectures Notes on Physics 151: Mechanics.” N.p., n.d. Web. 14 Jan. 2014. [3] Moser, Jurgen, and Eduard Zehnder. ”Transformation Theory.” Notes on Dynamical Systems . New York: Courant Institute of Mathematical Sciences, New York University, 2005. 14-25. PDF. [4] Tong, David. ”Lecture Notes on Hamiltonian Formalism.” N.p., n.d. Web. 14 Jan. 2014. [5] Torre, Charles. ”Lecture Notes on Infinitesimal Canonical Transformations.” N.p., n.d. Web. 14 Jan. 2014.
8
