2.6. PRINCIPLE OF HAMILTONIAN DYNAMICS (PHD) 97
introduced in the last subsection can be generalized to all physical fields. The most remark-
able characteristic of PHD is that the total energyHof the physical system is conserved:
(2.6.27)
d
dt
H(q(t),p(t)) = 0 ,where(q,p)are the solutions of the Hamiltonian system.
Now, we introduce the PHD.
Principle 2.45(Hamiltonian Dynamics).For any conservative physical system, there are two
sets of state functions
(2.6.28) u= (u 1 ,···,uN) and v= (v 1 ,···,vN),
such that the energy densityHis a function of (2.6.28):
(2.6.29) H=H(u,v,···,Dmu,Dmv), m≥ 0.
The total energy of the system is
(2.6.30) H=
∫ΩH(u,v,···,Dmu,Dmv)dx, Ω⊂R^3 ,provided that the system is described by continuous fields. Moreover the state functions u and
v satisfy the equations
(2.6.31)
∂u
∂t=αδH
δv,
∂v
∂t=−α
δH
δu,
whereαis a constant.
In general, the energy density (2.6.29) for a continuous field system depends onu,vonly
up to the first-order derivativesDuandDv:
(2.6.32) H=H(u,v,Du,Dv).
Then the Hamilton equations (2.6.31) can be expressed in the following form:
(2.6.33)
∂uk
∂t
=α[
−∂j(
∂H
∂ ζjk)
+
∂H
∂vk]
,
∂vk
∂t=α[
∂j(
∂H
∂ ξjk)
−
∂H
∂uk]
,
whereξjk,ζjkare variables corresponding to∂jukand∂jvk.
Hereafter, we always assume thatH is in the form (2.6.32). The following theorem
shows that the total energyHof the Hamiltonian system (2.6.31) (or (2.6.33)) is conserved.