14
1 Introduction
The properties of are reflected in the following properties of the Poisson bracket:
(i) { f, g + λh} = { f, g} + λ{ f, h},
(ii) { f, g} = −{g, f }, (bilinearit y and antisymmetr y o f ),
(iii) { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0, (closedness d = 0),
(iv) { f, g · h} = { f, g} · h + g · { f, h}, (derivation propert y (1.3.7)),
(v) I f { f, g} = 0 ∀g ∈ F(M) ⇒ f ≡ const. (nondegeneracy o f ).
It is not difficult to prove for the commutator of Hamiltonian vector fields:
[σ f , σ g ] = σ { f,g} .
(1.3.8)
1.3.6 In CM, all the observables are functions of points x ∈ M, hence locally can
be expressed as functions of a finite number 2n coordinate functions. In accordance
with the ‘philosophy’ of 1.2.6, we shall look for an interpretation of a finite number
of observables which contain systems of coordinate functions for a neighbourhood
of any point of M. This can be naturally done, if M is a homogeneous space of a
connected Lie group G (cf. [37, A.4]) corresponding to a group of empirical manipulations with objects relevant to the determination of the considered system. Since the
symplectic structure on M reflects important physical properties of many physical systems, it is desirable for the group action on M to conserve this structure. In
this case, one parameter subgroups of symmetries correspond to Hamiltonian flows
which can be physically interpreted.
1.3.7 From now on, we shall assume that (M; ) is a homogeneous space of a
connected Lie group G, on which the group G acts as an infinitely differentiable
group of symplectomorphisms F g (g ∈ G): F
∗
g = , F gh = F g ◦ F h (g, h, ∈ G),
and functions g → f (F g x) are in C
∞
(G, R) for all f ∈ F(M) and all x ∈ M. If
e ∈ G is the unit element of G, then F e := id M . To any ξ ∈ g (:= the Lie algebra of G)
there is a one-parameter group of symplectomorphisms t → F exp(tξ) of M generated
by the vector field σ ξ (compare with (1.3.1) ). If [ξ, η] denotes the commutator in g,
and [σ ξ , σ η ] ∈ X(M) the commutator of vector fields on M, then (see [1, Proposition
4.1.26])
[σ ξ , σ η ] = −σ [ξ,η] .
(1.3.9)
Every homogeneous symplectic manifold has universal covering symplectic
homogeneous manifold with respect to the universal covering group of G. On any
simply connected homogeneous symplectic manifold of a connected Lie group G,
the functions f ξ (ξ ∈ g) determined up to additive constants by the formula
i(σ ξ )) = −d f ξ
(1.3.10)
are defined globally on the manifold M, f ξ ∈ F(M), i.e. the vector fields σ ξ (ξ ∈ g)
are globally Hamiltonian. We shall assume that this is the case for our (M; ). Then
1 Introduction
The properties of are reflected in the following properties of the Poisson bracket:
(i) { f, g + λh} = { f, g} + λ{ f, h},
(ii) { f, g} = −{g, f }, (bilinearit y and antisymmetr y o f ),
(iii) { f, {g, h}} + {g, {h, f }} + {h, { f, g}} = 0, (closedness d = 0),
(iv) { f, g · h} = { f, g} · h + g · { f, h}, (derivation propert y (1.3.7)),
(v) I f { f, g} = 0 ∀g ∈ F(M) ⇒ f ≡ const. (nondegeneracy o f ).
It is not difficult to prove for the commutator of Hamiltonian vector fields:
[σ f , σ g ] = σ { f,g} .
(1.3.8)
1.3.6 In CM, all the observables are functions of points x ∈ M, hence locally can
be expressed as functions of a finite number 2n coordinate functions. In accordance
with the ‘philosophy’ of 1.2.6, we shall look for an interpretation of a finite number
of observables which contain systems of coordinate functions for a neighbourhood
of any point of M. This can be naturally done, if M is a homogeneous space of a
connected Lie group G (cf. [37, A.4]) corresponding to a group of empirical manipulations with objects relevant to the determination of the considered system. Since the
symplectic structure on M reflects important physical properties of many physical systems, it is desirable for the group action on M to conserve this structure. In
this case, one parameter subgroups of symmetries correspond to Hamiltonian flows
which can be physically interpreted.
1.3.7 From now on, we shall assume that (M; ) is a homogeneous space of a
connected Lie group G, on which the group G acts as an infinitely differentiable
group of symplectomorphisms F g (g ∈ G): F
∗
g = , F gh = F g ◦ F h (g, h, ∈ G),
and functions g → f (F g x) are in C
∞
(G, R) for all f ∈ F(M) and all x ∈ M. If
e ∈ G is the unit element of G, then F e := id M . To any ξ ∈ g (:= the Lie algebra of G)
there is a one-parameter group of symplectomorphisms t → F exp(tξ) of M generated
by the vector field σ ξ (compare with (1.3.1) ). If [ξ, η] denotes the commutator in g,
and [σ ξ , σ η ] ∈ X(M) the commutator of vector fields on M, then (see [1, Proposition
4.1.26])
[σ ξ , σ η ] = −σ [ξ,η] .
(1.3.9)
Every homogeneous symplectic manifold has universal covering symplectic
homogeneous manifold with respect to the universal covering group of G. On any
simply connected homogeneous symplectic manifold of a connected Lie group G,
the functions f ξ (ξ ∈ g) determined up to additive constants by the formula
i(σ ξ )) = −d f ξ
(1.3.10)
are defined globally on the manifold M, f ξ ∈ F(M), i.e. the vector fields σ ξ (ξ ∈ g)
are globally Hamiltonian. We shall assume that this is the case for our (M; ). Then