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Definite integrals and Functionals

  1. Jul 8, 2017 #1
    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. jcsd
  3. Jul 9, 2017 #2
    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
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