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7.5 Lagrange Brackets

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7.5 Lagrange Brackets Chapter 7 Brackets Module 1 Poisson brackets and Lagrange brackets 1 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) i1 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 ofFF12, , 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 2 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 qqiippjj qpi,. j ik jk ij kkqk 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 ti1 qii p FFHFHn (using Hamilton's equations) ti1 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 Qi Q i( q1 , q 2 ,..., q n ; p 1 , p 2 ,..., p n ; t ) (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) qijj11 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 ]. qPijqQij FFFFFnn QP qq 1 1jj 1 1ii 1 (7.7) pijj11 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 ]. pPijpQji n FFFF1 2 1 2 FF12, i1 qi p i p i q i n FFFFFFFFp p q q 1i 2 1 i 2 1 i 2 1 i 2 ij,1 QPpjji PQp jji QPq jji PQq jji nnFFFF pq nn FFpq 1 2ii 2 1 22ii Q p P q P P p Q q Q ji11j i j i jji11 j i j i j n FFFF1 2 1 2 FF12,. QPPQ j1 j j j j Thus the theorem is proved. 5 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) i1 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) iiFFFFFF1 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. 6 .
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