Chapter 1
Classical Mechanics
We assume that the reader is familiar with the basic notions from the theory
of smooth (that is,
C∞)
manifolds and recall here the standard notation. Un-
less it is stated explicitly otherwise, all maps are assumed to be smooth and
all functions are assumed to be smooth and real-valued. Local coordinates
q =
(q1,
. . . ,
qn)
on a smooth n-dimensional manifold M at a point q M
are Cartesian coordinates on ϕ(U) Rn, where (U, ϕ) is a coordinate chart
on M centered at q U. For f : U
Rn
we denote (f
ϕ−1)(q1,
. . . ,
qn)
by
f(q), and we let
∂f
∂q
=
∂f
∂q1
, . . . ,
∂f
∂qn
stand for the gradient of a function f at a point q
Rn
with Cartesian
coordinates
(q1,
. . . ,
qn).
We denote by
A•(M)
=
n
k=0
Ak(M)
the graded algebra of smooth differential forms on M with respect to the
wedge product, and by d the de Rham differential a graded derivation of
A•(M)
of degree 1 such that df is a differential of a function f
A0(M)
=
C∞(M).
Let Vect(M) be the Lie algebra of smooth vector fields on M with
the Lie bracket [ , ], given by a commutator of vector fields. For X Vect(M)
we denote by LX and iX , respectively, the Lie derivative along X and the
inner product with X. The Lie derivative is a degree 0 derivation of
A•(M)
which commutes with d and satisfies LX(f) = X(f) for f
A0(M),
and
the inner product is a degree −1 derivation of
A•(M)
satisfying iX (f) = 0
3
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