So this is, say, indexed.

This is number 0, this is -1, this is 1 and so forth to infinity.

2, etc.- 2, etc.

So we have such a collection of particles

each one of them having mass m and

the springs having the hook index k.

Hook index k.

It is not hard to find that the action for

this system, which is just the integral

over dt from t1 to t of the Lagrangian which

is just the difference between total

kinetic energy- total potential energy.

And in our case, this is just t1 t2.

Where here, total kinetic energy is just the sum of

all kinetic energies of the particles from minus infinity to plus infinity.

m phi i dot squared over 2,

whose phi I will explain in a second.

Sum over i from- infinity

to + infinity k phi i + 1-

phi i squared over 2.

Phi is just displacement.

So in equilibrium particles positioned at

equal dispositions from each other, along this line.

So but when we move one of the particles in one or

the other direction from its equilibrium, the displacement from

the equilibrium we call as phi i, where i is the number of the particle.

So the displacement if we moved one, or they started to shake somehow.

There are waves running.

We're going to describe this wave with this discussion which we are going

to conduct.

So they are changing in time.

So that's what we get.

And now, it's not hard to see.

I assume that listeners of this course, they know the minimal action principle.

At least in non relativistic situation, and minimal action principle in

non relativistic situation leads from this action to the falling equation.

m phi i double dot is =

k phi i +1- phi i-

k phi i- phi i- 1.

So the meaning of this formula is very simple.

It's a just Newton's second law, m times acceleration of e's

particle is equal to the force acting on it from one side,

from this side, by the spring which is on the right hand side.

And the force by the spring on the left hand side.