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CHAPTER 3

Supersymmetric Classical
Mechanics

3.1 Classical Poisson Bracket, its General-
izations
In classical mechanics we encounter the notion of Poisson brackets in
connection with transformations of the generalized coordinates and
generalizaed momenta that leave the form of Hamilton™s equations of
motion unchanged [1-3]. Such transformations are called canonical
and the main property of the Poisson bracket is its invariance with
respect to the canonical transformations. In terms of the generalized
coordinates q1 , q2 , . . . qn and generalized momenta p1 , p2 , . . . , pn the
Poisson bracket in classical mechanics is de¬ned by
n
‚f ‚g ‚f ‚g
{f, g} = ’ (3.1)
‚qj ‚pj ‚pj ‚qj
j=1

for any pair of functions f ≡ f (q1 , q2 , . . . qn ; p1 , p2 , . . . pn ; t) and g ≡
g(q1 , q2 , . . . qn ; p1 , p2 , . . . pn ; t).
Recall that whereas the Lagrangian in classical mechanics is
known in terms of the generalized coordinates (qi ), the generalized ve-
locities (qi ), and time (t), namely L = L (q1 , q2 , . . . qn ; q1 , q2 , . . . qn , t),
™ ™™ ™
the corresponding Hamiltonian is given in terms of the generalized
coordinates (qi ), generalized momenta (pi ), and time (t), namely


© 2001 by Chapman & Hall/CRC
‚L
H = H (q1 , q2 , . . . qn ; p1 , p2 , . . . pn ; t) where pi = ‚ qi , i = 1, 2, . . . n.

The relationship between the Lagrangian and Hamiltonian is pro-
n
vided by the Legendre transformation H = pi qi ’ L and Hamil-

i=1
ton™s canonical equations of motion are obtained by varying both
sides of it
‚H
= qi

‚pi
‚H
= ’pi

‚qi
‚H ‚L
=’ (3.2)
‚t ‚t
Relations (3.2) prescribe a set of 2n ¬rst-order di¬erential equa-
tions for the 2n variables (qi , pi ). In contrast Lagrange™s equations
involve n second-order di¬erential equations for the n generalized
coordinates
d ‚L ‚L
= i = 1, 2, . . . n (3.3)
dt ‚ qi
™ ‚qi
For a given transformation (qi , pi ) ’ (Qi , Pi ) to be canonical we
need to have

{Qi , Qj } = 0, {Pi , Pj } = 0,
{Qi , Pj } = δij (3.4)

These conditions are both necessary and su¬cient. Often (3.4)
is used as a de¬nition for the canonically conjugate coordinates and
momenta. Some obvious properties of the Poisson brackets are

anti-symmetry: {f, g} = ’{g, f }, {f, c} = 0 (3.5a)
linearity: {f1 + f2 , g} = {f1 , g} + {f2 , g} (3.5b)
chain-rule: {f1 f2 , g} = f1 {f2 , g} + {f1 , g}f2 (3.5c)
Jacobi identity: {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0 (3.5d)

where c is a constant and the functions involved are known in terms
of generalized coordinates, momenta, and time.
The transition from classical to quantum mechanics is formu-
lated in terms of the commutators from the classical Poisson bracket


© 2001 by Chapman & Hall/CRC
relations. Indeed it can be readily checked that the commutator
of two operators satis¬es all the properties of the Poisson bracket
summarized in (3.5). The underlying fundamental commutation re-
lation in quantum mechanics being [x, p] = i¯ , the classical Poisson
h
bracket may be viewed as an outcome of the following limit on the
commutator
lim f , g
= {f, g} (3.6)
h
¯ ’ 0 i¯ h
where f , g stands for the commutator of the two operators f and

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