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

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