1.3 Classical Hamiltonian Mechanics
15
arbitrary additive constants in the definitions of f ξ ’s can be chosen such that the
mapping ξ → f ξ from g to F(M) will be linear. Then
{ f ξ , f η } = − f [ξ,η] + C(ξ, η),
(1.3.11)
where C is a bilinear antisymmetric mapping from g × g to real constants on M called
a two-cocycle on g with values in R. Any change of constants in f ξ ’s (conserving the
linearity of ξ → f ξ ) leads to an equivalent cocycle C
(ξ, η) = C(ξ, η) + a([ξ, η]),
where a ∈ g
∗ (:= the dual of g). Equivalence classes of two-cocycles form the commutative (additive)
2-cohomology group H
2
(g, R)
of g with values in R. This group is isomorphic to H
2
(G, S
1
) if G is simply connected
(compare [321, Chap. 10.4.]). This isomorphism determines a canonical bijection
between classes of irreducible projective representations and symplectic transitive
actions of a simply connected Lie group. This bijection associates the class of all
representations corresponding to the given (similarity class of a) multiplier with the
class of symplectic actions with the corresponding (equivalence class of a) cocycle,
compare also [28, 139]. If the multiplier m corresponds to the cocycle C from
(1.3.11), then the central extension G m of G (cf. [174, 15.2, Thm. 1]) acts on M in
such a way, that
{ f ξ , f η } = − f [ξ,η] for all ξ, η ∈ g m ,
(1.3.12)
if the added vector fields act on M trivially and constants in f ξ ’s are properly chosen.
If the action of G on M satisfies (1.3.11) with C ≡ 0, then it is called a Poisson
action [7], and the symplectic manifold M is called exactly homogeneous [174].
1.3.8 Any observable f ∈ F(M) on the homogeneous symplectic manifold M with
globally defined Hamiltonian functions f ξ (ξ ∈ g) can be expressed as a function of
the ‘basic observables f ξ ’. Hence measurement of any f ∈ F(M) can be reduced to
the measurements of f ξ ’s. This does not make easier, however, of an ascribing a direct
physical (i.e. empirical) interpretation to an arbitrary f ∈ F(M) and the situation is
similar to that one of QM, see 1.2.8.
1.3.9 A time evolution on (M; ) is defined in CM as a differentiable one-parameter
group of symplectomorphisms with a globally defined Hamiltonian function h ∈
F(M). This one-parameter group might be either a subgroup of G, or it is separately
defined. In each case the group G might contain an invariance subgroup of h—the
symmetry group of the dynamics (determining integrals of motion—conservation
laws).
15
arbitrary additive constants in the definitions of f ξ ’s can be chosen such that the
mapping ξ → f ξ from g to F(M) will be linear. Then
{ f ξ , f η } = − f [ξ,η] + C(ξ, η),
(1.3.11)
where C is a bilinear antisymmetric mapping from g × g to real constants on M called
a two-cocycle on g with values in R. Any change of constants in f ξ ’s (conserving the
linearity of ξ → f ξ ) leads to an equivalent cocycle C
(ξ, η) = C(ξ, η) + a([ξ, η]),
where a ∈ g
∗ (:= the dual of g). Equivalence classes of two-cocycles form the commutative (additive)
2-cohomology group H
2
(g, R)
of g with values in R. This group is isomorphic to H
2
(G, S
1
) if G is simply connected
(compare [321, Chap. 10.4.]). This isomorphism determines a canonical bijection
between classes of irreducible projective representations and symplectic transitive
actions of a simply connected Lie group. This bijection associates the class of all
representations corresponding to the given (similarity class of a) multiplier with the
class of symplectic actions with the corresponding (equivalence class of a) cocycle,
compare also [28, 139]. If the multiplier m corresponds to the cocycle C from
(1.3.11), then the central extension G m of G (cf. [174, 15.2, Thm. 1]) acts on M in
such a way, that
{ f ξ , f η } = − f [ξ,η] for all ξ, η ∈ g m ,
(1.3.12)
if the added vector fields act on M trivially and constants in f ξ ’s are properly chosen.
If the action of G on M satisfies (1.3.11) with C ≡ 0, then it is called a Poisson
action [7], and the symplectic manifold M is called exactly homogeneous [174].
1.3.8 Any observable f ∈ F(M) on the homogeneous symplectic manifold M with
globally defined Hamiltonian functions f ξ (ξ ∈ g) can be expressed as a function of
the ‘basic observables f ξ ’. Hence measurement of any f ∈ F(M) can be reduced to
the measurements of f ξ ’s. This does not make easier, however, of an ascribing a direct
physical (i.e. empirical) interpretation to an arbitrary f ∈ F(M) and the situation is
similar to that one of QM, see 1.2.8.
1.3.9 A time evolution on (M; ) is defined in CM as a differentiable one-parameter
group of symplectomorphisms with a globally defined Hamiltonian function h ∈
F(M). This one-parameter group might be either a subgroup of G, or it is separately
defined. In each case the group G might contain an invariance subgroup of h—the
symmetry group of the dynamics (determining integrals of motion—conservation
laws).