Why does the action vary to zero with respect to the field?

[georg gill]

http://en.wikipedia.org/wiki/Lagrangian#Explanation

I am trying to prove pV=nRT and in order to do so one need to get lagrangian (not the math formula it seems)

Here is an explanation

http://en.wikipedia.org/wiki/Lagrangian#Explanation

why is

$$\frac{\delta S}{\delta \varphi _i}=0$$?

S is a point given in time and space but I guess my problem is what is
$$\varphi$$



I guess that it is the value of the field at that point in spacetime as they write does not help me much to get what it is

 

[strangerep]

I am trying to prove pV=nRT [...]

You didn't say what these symbols mean.

why is

$$\frac{\delta S}{\delta \varphi _i}=0$$?

S is a point given in time and space

No. S is the action. Read a bit further on that Wiki page. It says
$$\mathcal{S} [\varphi_i] = \int{\mathcal{L} [\varphi_i (x)]\, \mathrm{d}^4x}$$
Lagrangian/Hamiltonian mechanics start from a principle of least action, meaning
that the total action is assumed not to vary under small variations of the generalized coordinates (i.e., the $\varphi_i$ in this case) and the equations of motion are
then derived from this principle.

but I guess my problem is what is $\varphi$

It's a generalized configuration variable.

BTW, this question probably belongs over on the classical mechanics forum. It sounds like you really need a textbook, and someone over there could probably suggest one.