Mathematical Principles of Theoretical Physics

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.