Nonlinear Mechanics - Physics at Oregon State University

3.1. NOTATION 35
Those of you who have studied special relativity should find (??) congenial.
Remember the definition of a Lorentz transformation: any 4×4 matrix
Λ that satisfies
g = Λ · g · Λ T
(3.11)
is a Lorentz transformation. 1 The matrix
⎛
1 0 0
⎞
0
⎜
g = ⎜ 0
⎝ 0
−1
0
0
−1
0 ⎟
0 ⎠
0 0 0 −1
(3.12)
is called the metric or metric tensor. Forgive me for exaggerating slightly:
everything there is to know about special relativity flows out of (3.11). We
say that Lorentz transformations “preserve the metric,” i.e. leave the metric
invariant. The geometry of space and time is encapsulated in (12). By the
same token, canonical transformations preserve the metric J. The geometry
of phase space is encapsulated in the definition of J. Since J is symplectic,
canonical transformations are symplectic transformation, they preserve the
symplectic metric.
Equation (4.10) is the starting point for the modern approach to mechanics
that uses the tools of Lie group theory. I will only mention in passing
some points of contact with group theory. Both Goldstein’s and Schenk’s
texts have much more on the subject.
3.1.1 Poisson Brackets
Equation (3.10) is really shorthand for four equations, e.g.
∂Q ∂P
∂q ∂p
∂P ∂Q
− = 1 (3.13)
∂q ∂p
This combination of derivatives is called a Poisson bracket. The usual notation
is
∂X ∂Y ∂X ∂Y
− ≡ [X, Y ]q,p
∂q ∂p ∂q ∂p
(3.14)
The quantity on the left is called a Poisson bracket. Then (3.13) becomes
[Q, P ]q,p = 1 (3.15)
1 It is not a good idea to use matrix notation in relativity because of the ambiguity
inherent in covariant and contravariant indices. Normally one would write (11) using
tensor notation.