Poisson brackets

The differential expression

$$ \tag{1 } ( u , v ) = \sum_{i=1} ^ { n } \left ( \frac{\partial u }{\partial q _ {i} } \frac{\partial v }{\partial p _ {i} } - \frac{\partial u }{\partial p _ {i} } \frac{\partial v }{\partial q _ {i} } \right ) , $$

depending on two functions $ u ( q , p ) $ and $ v ( q , p ) $ of $ 2n $ variables $ q = ( q _ {1} \dots q _ {n} ) $, $ p = ( p _ {1} \dots p _ {n} ) $. The Poisson brackets, introduced by S. Poisson [1], are a particular case of the Jacobi brackets. The Poisson brackets are a bilinear form in the functions $ u $ and $ v $, such that

$$ ( u , v ) = - ( v , u ) $$

and the Jacobi identity

$$ ( u , ( v , w ) ) + ( v , ( w , u ) ) + ( w , ( u , v ) ) = 0 $$

holds (see [2]).

The Poisson brackets are used in the theory of first-order partial differential equations and are a useful mathematical tool in analytical mechanics (see [3]–[5]). For example, if $ q $ and $ p $ are canonical variables and a transformation

$$ \tag{2 } Q = Q ( q , p ) ,\ \ P = P ( q , p ) $$

is given, where $ Q = ( Q _ {1} \dots Q _ {n} ) $, $ P = ( P _ {1} \dots P _ {n} ) $ and the $ ( n \times n ) $- matrices

$$ \tag{3 } ( P , P ) ,\ ( Q , Q ) ,\ ( Q , P ) $$

are constructed with entries $ ( P _ {i} , P _ {j} ) $, $ ( Q _ {i} , Q _ {j} ) $, $ ( Q _ {i} , P _ {j} ) $, respectively, then (2) is a canonical transformation if and only if the first two matrices in (3) are zero and the third is the unit matrix.

The Poisson brackets, computed for the case when $ u $ and $ v $ are replaced in (1) by some pair of coordinate functions in $ q $ and $ p $, are also called fundamental brackets.

References

[1]	S. Poisson, J. Ecole Polytechn. , 8 (1809) pp. 266–344
[2]	C.G.J. Jacobi, "Nova methodus, aequationes differentiales partiales primi ordinis inter numurum variabilium quemcunque propositas integrandi" J. Reine Angew. Math. , 60 (1862) pp. 1–181
[3]	E.T. Whittaker, "Analytical dynamics of particles and rigid bodies" , Dover, reprint (1944)
[4]	A.I. Lur'e, "Analytical mechanics" , Moscow (1961) (In Russian)
[5]	H. Goldstein, "Classical mechanics" , Addison-Wesley (1957)

Comments

Other basic properties of Poisson brackets are invariance under canonical transformations and the fact that $ ( F, H) $ expresses the derivative of $ F( q, p) $ along trajectories, if $ H $ is the Hamiltonian, so that the corresponding Hamiltonian equations are $ \dot{q} _ {i} = ( q _ {i} , H) $, $ \dot{p} _ {i} =( p _ {i} , H) $, which for a "standard" Hamiltonian of the form $ H=( \sum p _ {i} ^ {2} )/2+ V( q) $ gives back $ \dot{q} _ {i} = p _ {i} $, $ \dot{p} _ {i} = - \partial H/ \partial q _ {i} $. Therefore $ ( F, H) $ expresses a conservation law, i.e. $ F $ is a conserved quantity.

The Poisson brackets may be defined for functionals depending on a function $ q( x) $, as

$$ F[ q] = \int\limits _ {- \infty } ^ \infty \widetilde{F} ( q ,q ^ {(1)} , q ^ {(2)} ,\dots) dx, $$

with $ q ^ {(n)} = d ^ {n} q/dx ^ {n} $.

One has

$$ ( F, G) = \int\limits _ {- \infty } ^ \infty \frac{\delta \widetilde{F} }{\delta q } \frac{d}{dx} \frac{\delta \widetilde{G} }{\delta q } dx, $$

with $ {\delta \widetilde{F} } / {\delta q } $, $ {\delta \widetilde{G} } / {\delta q } $ variational derivatives, i.e.

$$ \frac{\delta \widetilde{F} }{\delta q } = \sum \left ( - \frac{d}{dx} \right ) ^ {n} \frac{\partial \widetilde{F} }{\partial q ^ {(n)} } . $$

References

[a1]	A.C. Newell, "Solitons in mathematical physics" , SIAM (1985)
[a2]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a3]	R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin (1978)
[a4]	F.R. [F.R. Gantmakher] Gantmacher, "Lectures in analytical mechanics" , MIR (1975) (Translated from Russian)