# Definite integrals and Functionals

1. Jul 8, 2017

### jamie.j1989

Taken from Emmy Noether's wonderful theorem by Dwight. E Neuenschwander. Page 28

1. The problem statement, all variables and given/known data

Under what circumstances are these definite integrals functionals;
a) Mechanical work as a particle moves from position a to position b, while acted upon by a force F.

$$W=\int_a^b\boldsymbol{F}\bullet d\boldsymbol{r},\qquad (1)$$

b) The Entropy change $\Delta S$, in terms of heat $dQ$ added to a system at absolute temperature T, for a change of thermodynamic state from a to b.

$$\Delta S=\int_a^b\frac{dQ}{T},\qquad (2)$$

2. Relevant equations
A functional $\Gamma$ is a mapping of a well defined set of functions onto the real numbers. And is given by the definite integral

$$\Gamma=\int_a^bL(q^\mu,\dot{q}^\mu,t)dt,\qquad (3)$$
Where L is the Lagrangian of the functional and the label $\mu$ on the generalised coordinates $q$ distinguishes between N dependent variables.

3. The attempt at a solution
For a). From the above definition of $\Gamma$ we can compare (1) and (3), if the force F is compared to the Lagrangian in the functional then it needs to be a function of the independent variable r ?

And similarly for b), if the absolute temperature of the system in (2) is a function of the heat then the definite integral is a functional?

Is it an issue if they aren't functions of a dependent variable $q$ and it's first derivative $q'$ with respect to the independent variable?

2. Jul 9, 2017

### Stavros Kiri

In general I don't think so, but it would be hard to get Euler-Lagrange equations then, if the form is not simple.
The definition of 'functional' is quite general. E.g. see:
https://en.m.wikipedia.org/wiki/Functional_(mathematics)

Last edited: Jul 9, 2017