2.5. PRINCIPLE OF LAGRANGIAN DYNAMICS (PLD) 81
The total energyHof theN-body system is given by
(2.5.7) H(x,y) =
N
∑
k= 1
1
2 mk
y^2 k+V(x),
whereykis the momentum of thek-th planet, andV(x)is the potential energy as in (2.5.3).
Then the motion equations derived from the PHD are as follows
(2.5.8)
∂xk
∂t
=
∂H
∂yk
,
∂yk
∂t
=−
∂H
∂xk
.
whereH(x,y)is defined by (2.5.7). It is easy to check that the equations (2.5.8) are equivalent
to those in (2.5.1)-(2.5.2).
2.5.2 Elastic waves
In an elastic continuous medium, the wave vibration is described by the PLD.
LetΩ⊂R^3 be a domain of elastic continuous medium, and the functionu(x,t)represent
the displacement of the medium at timetandx∈Ω. Then the total kinetic energy is
(2.5.9) T=
∫
Ω
1
2
ρ
∣
∣
∣
∣
∂u
∂t
∣
∣
∣
∣
2
dx,
whereρis the density of the medium.
For an elastic material, the deformation potential energyVis taken in the general form
(2.5.10) V=
∫
Ω
[
1
2
k|∇u|^2 +F(x,u)
]
dx,
wherekis a constant.
The Lagrange action for the elastic wave is written as
(2.5.11) L=
∫t 1
t 0
(T−V)dt=
∫t 1
t 0
∫
Ω
[
1
2
ρu ̇^2 −
1
2
k|∇u|^2 −F(x,u)
]
dxdt.
By the PLD, the wave equation is derived by
δL= 0 ,
and it follows from (2.5.11) that
(2.5.12) ρ
∂^2 u
∂t^2
−k∆u=f(x,u) forx∈Ω.
where
f(x,z) =
∂F(x,z)
∂z
.
The equation (2.5.12) is the usual wave equation describing elastic vibration ina continuous
medium.