7.5 Lagrange Brackets

Chapter 7

Brackets

Module 1

Poisson brackets and Lagrange brackets

Brackets are a powerful and sophisticated tool in Classical mechanics particularly in Hamiltonian formalism. It is a way of characterization of canonical transformations by using an operation, known as bracket. It is very useful in mathematical formulation of classical mechanics. It is a powerful geometrical structure on phase space. Basically it is a mapping which maps two functions into a new function. Reformulation of classical mechanics is indeed possible in terms of brackets. It provides a direct link between classical and quantum mechanics.

7.1 Poisson brackets Poisson bracket is an operation which takes two functions of phase space and time and produces a new function.

Let us consider two functions F1(,,) qii p t and F2 (,,) qii p t . The Poisson bracket is defined as

n FFFF1  2  1  2 FF12,.  (7.1) i1 qi  p i  p i  q i

For single degree of freedom, Poisson bracket takes the following form

FFFF    FF,. 1 2 1 2 12 q  p  p  q

Basically in Poisson bracket, each term contains one derivative of F1 and one derivative of F2. One derivative is considered with respect to a coordinate q and the other is with respect to the conjugate momentum p. A change of sign occurs in the terms depending on which function is differentiated with respect to the coordinate and which with respect to the momentum.

Remark: Some authors use square brackets to represent the Poisson bracket i.e. instead ofFF12, , they prefer to use FF12,  .

Example 1: Let F12( q , p ) p  q , F ( q , p )  sin q . Then the Poisson bracket is given by

FFFF    F, F 1 2  1 2  (  1)(0)  (1)(cos q )   cos q . 12 q  p  p  q

Example 2: If F12( q , p ) p  q , F ( q , p )  cos p then the Poisson bracket is

FFFF    F, F 1 2  1 2  (1)(  sin p )  (1)(0)   sin p . 12 q  p  p  q

Example 3: For F12( q , p ) q , F ( q , p )  p  2 q then the Poisson bracket is

FFFF    FF, 1 2  1 2  (1)(1)  (0)(  2)  1. 12 q  p  p  q

Example 5: Poisson brackets of the canonical variables have the following form

1, for ij qi, q j  0, p i , p j  0, q i , p j   ij where ij   0, for ij

Now,

qq q qqiijj j qi qqij,    0, (since  0 ,  0 ) k qk  p k  p k  q k pk pk

pp ppiijj pi p j ppij,    0, (since  0 ,  0 ) k qk  p k  p k  q k qk qk

qqiippjj qpi,. j    ik  jk   ij kkqk  p k  p k  q k

p qi j (since  ik ,   jk ). qk pk 3

7.2 Poisson theorem

The total time rate of evaluation of any dynamical variable F(,,) q p t is given by dF F FH,  , where H is the Hamiltonian of the system. dt t Proof:

dF Fn   F  F   qpii  dt ti1   qii  p FFHFHn         (using Hamilton's equations) ti1   qi  p i  p i  q i F FH,. t (7.2)

7.3 Hamilton’s equations of motion in terms of Poisson’s brackets

We know that Hamilton’s equations of motion are given by

HH qp,   . pq

q  H  q  H  H Now, q, H     q q  p  p  q  p

p  H  p  H  H & p, H      p . q  p  p  q  q

7.4 Theorem: Poisson bracket is invariant under canonical transformation.

Proof: Let us consider the canonical transformation (,,)(,,)q p t Q P t given by

Q Q( q , q ,..., q ; p , p ,..., p ; t ) i i1 2 n 1 2 n (7.3) Pi P i( q1 , q 2 ,..., q n ; p 1 , p 2 ,..., p n ; t ) 4

and let the dynamical variables F1(,,) q p t and F2 (,,) q p t transform to F1(,,) q p t and

F2 (,,) q p t given by

Fqpt1(,,) FqQPt 1 (,,),(,,), pQPtt FQPt 1 (,,) (7.4)

Fqpt2(,,) FqQPt 2 (,,),(,,), pQPtt FQPt 2 (,,) . (7.5)

We have to show that FFFF1,,. 2   1 2

Now,

FFFFFnn QP    pp   1 1jj  1  1ii  1 (7.6)     qijj11  Q j  q i  P j  q i    Q j  P j  P j  Q j 

Q p P p [since by (6.24) of canonical transformations we have j  i & j  i ]. qPijqQij

FFFFFnn QP    qq   1 1jj  1   1ii  1 (7.7)     pijj11  Q j  p i  P j  p i    Q j  P j  P j  Q j 

Q q P q [since by (6.24) of canonical transformations we have j  i & i  j ]. pPijpQji

n FFFF1  2  1  2 FF12,    i1 qi  p i  p i  q i n FFFFFFFFp    p    q    q  1i 2  1 i 2  1 i 2  1 i 2   ij,1 QPpjji  PQp jji  QPq jji  PQq jji nnFFFF pq nn  FFpq 1 2ii  2  1 22ii Q  p  P  q  P  P p  Q  q  Q ji11j i j i jji11 j i j i j n  FFFF1  2  1  2    FF12,. QPPQ    j1 j j j j Thus the theorem is proved.

7.5 Lagrange brackets

For the purposes of mathematical formulation of classical mechanics, Joseph Louis Lagrange introduced Lagrange brackets in 1808–1810.

If (q1 , q 2 ,..., qnn ; p 1 , p 2 ,..., p ) is a system of canonical coordinates in a phase space and if

FF12, be the functions of those variables then Lagrange bracket is defined as

n qi  p i  p i  q i FF12,.  (7.8) i1 FFFF1  2  1  2

7.6 Properties of Lagrange brackets

(i) Lagrange brackets are anticommutative i.e. FFFF1,, 2  2 1  . (ii) Lagrange bracket does not depend on the canonical coordinates i.e. Lagrange bracket is an invariant under the canonical transformations.

7.7 Lagrange bracket is canonical invariant

Lagrange bracket of FF12, is defined as

qi  p i  q i  p i (,) q i p i FF12,     (7.9) iiFFFFFF1  2  2  1 (,) 1 2

(,)qp Since  iiis invariant under canonical transformation [by the result (6.32)] so i (,)FF12 Lagrange’s bracket is invariant under canonical transformation.