Dummy Index and Free Index

When summation is made on a large number of particles then the variable used to indicate the large no of particles is called Dummy Index if it can be changed or replaced with another variable without any change in the system. 
e.g; `x_{1},x_{2},x_{3},.........x_{1}`
=`\sum_i^nx_{i}`

when summation is made on a large numbers of particles then the variable used indicate the large no. of particles is called Free index. If it appears only once in each additive term.
e.g. consider a matrix m`\times`n its first row is represented as
`a_{ij}=a_{11}+a_{12}+a_{13}+......+a_{n}` here i=1 is free index and j=1,2,3 is dummy index.

Mechanics of system of particles

In mechanics of a single particle, we study only the external force while in case of system of particles we study both external and internal forces.
If we discuss the action of free or anyother phenomenon for a number of particles which are in combination then this combination of particles form a system of particles.