Mathematical Principles of Theoretical Physics

80 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS

In classical mechanics, a physical motion system can be described by three dynamical
principles: the Newtonian Dynamics, the Lagrangian Dynamics, and the Hamiltonian Dy-
namics. Both the Lagrangian dynamics and the Hamiltonian dynamics remain valid in other
physical fields such as the electrodynamics and quantum physics.
The three dynamical principles are equivalent in describing the motion of anN-body
system. Consider anN-body system of planets, with massesm 1 ,···,mNand coordinates
xk= (x^1 k,x^2 k,x^3 k).

1.Newtonian dynamics.The motion equations governing theNplanets are the Newtonian
second law:

(2.5.1) mk

d^2 xk

dt^2

=Fk for 1≤k≤N,

whereFkis the gravitational force acting on thek-th planet by the other planets, which can
be expressed as

(2.5.2) Fk=−∑
j 6 =k

mkmjG

|xj−xk|^3

(xj−xk),

whereGis the gravitational constant.

2.Lagrangian dynamics. Based on the least action principle, the Lagrange densityLof
theN-body system is
L=T−V,

whereTis the total kinetic energy, andVis the potential energy:

(2.5.3) T=

1

2

N

∑

k= 1

mk

(

dxk

dt

) 2

, V=− ∑

i,j= 1 ,i 6 =j

mimjG

2 |xi−xj|

,

Hence, the Lagrange action is

(2.5.4) L=

∫t 1

t 0

Ldt=

1

2

∫t 1

t 0

[

N

∑

k= 1

mkx ̇^2 k+∑

i 6 =j

mimjG

|xi−xj|

]

dt.

The variational derivative operatorδLofLis

(2.5.5) δL=−mkx ̈k−∑
j 6 =k

mkmjG

|xj−xk|^3

(xj−xk),

and the motion equations of the Lagrangian dynamics are given by

(2.5.6) δL= 0.

It is clear that the motion equations (2.5.6) derived from the Lagrangian dynamics is the
same as the equations (2.5.1)-(2.5.2) of the Newtonian dynamics.

3.Hamiltonian dynamics. In the next section we shall introduce the principle of Hamil-
tonian dynamics (PHD), and we derive here the Hamiltonian system for theN-body motion.