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the homogeneity (shift symmetry) of space (position in space is the
canonical conjugate quantity to momentum). That is, conservation of
momentum is equivalent to the fact that physical laws do not depend on
position.

Consider closed system that consists of n particles which move
with different velocities and interact with themselves (fig.5.1.). For
example, on a particle under a number 1 acts particle number 2 with

force F and vice versa. So we can write n- equations of Newton's
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second law for all particle of this system

    
)
...  F  d( m v )  (5.1)
 F
F
 F
12 13 14 n 1 dt  

    d( m v ) 
F  F  F ...  F  2 2
21 23 24 2 n 
dt

    d( m v ) 
F 31  F 32  F 34 ...  F n 3  3 3 

dt 
.......... .......... .......... .......... .......... ......  

   d( m n (  n)1 v n 1 
F n (  1)1  F n (  2)1 ...  F n (  n)1  dt 

   d( m v ) 
F n1  F n2  .....  F n( n  )1  n n 

dt 
Add these equations and take into account Newton's third law
     
F   F ; F   F ........F n n 1    F n 1 n , (5.2)
31
21
13
12
we obtain
d   
 vm 1 1  m 2 v 2   m n v n  0 . (5.3)
dt

It's possible, when value under differential is constant
  
m 1 v  m 2 v   m n v  const . (5.4)
n
2
1
Or
n 
 i  const . (5.5)
p
i n

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