2.3 Integrable Systems
17
{I i , I j } P oisson = 0,
(2.14)
for i = 1, . . . , N and j = 1, . . . , N. Then the quantities I i form a
set of N phase space coordinates. In conservative systems, the Hamiltonian,
H (p 1 , . . . , p N , q 1 , . . . , q N ), will be one of the constants of the motion. In general,
the equation of motion of a phase function, f = f (p 1 , . . . , p N , q 1 , . . . , q N , t), is
given by
df
dt
=
∂f
∂t
+ {H, f } P oisson .
(2.15)
Thus Eqs. (2.14) and (2.15) imply that
dI i
dt = 0. If a system is integrable, there are
no internal nonlinear resonances leading to chaos. All orbits lie on N-dimensional
surfaces in the 2N -dimensional phase space.
2.3.1 Noether’s Theorem
As was shown by Noether (1918), isolating integrals result from symmetries. For
example, the total energy is an isolating integral (is a constant of the motion) for
systems that are homogeneous in time (invariant under a translation in time). Total
angular momentum is an isolating integral for systems that are isotropic in space.
Noether’s theorem is generally formulated in terms of the Lagrangian (see Goldstein 1980 and Appendix A). Let us consider a dynamical system with N degrees
of freedom whose state is given by the set of generalized velocities and positions
({ ˙
q i }, {q i }). Let us consider a system whose Lagrangian, L = L({ ˙
q i }, {q i }), is
known. For simplicity, we consider a system with a time-independent Lagrangian.
The equations of motion are given by the Lagrange equations
∂L
∂q i
−
d
dt
∂L
∂ ˙
q i
= 0, (i = 1, . . . , N).
(2.16)
For such systems, Noether’s theorem may be stated as follows.
• Noether’s Theorem If a transformation
t → t
= t + δt, q i (t) → q
i (t
) = q i (t) + δq i (t), and
˙
q i → ˙
q
i (t
) = ˙
q i (t) + δ ˙
q i (t)
(for i = 1, . . . , N) leaves the Lagrangian form invariant,
L({ ˙
q i (t)}, {q i (t)}) → L
({ ˙
q
i (t
)}, {q
i (t
)}) = L({ ˙
q
i (t
)}, {q
i (t
)}),
(2.17)