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jamie.j1989
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Taken from Emmy Noether's wonderful theorem by Dwight. E Neuenschwander. Page 28
1. Homework Statement
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.
$$\Delta S=\int_a^b\frac{dQ}{T},\qquad (2)$$
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.
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?
1. Homework Statement
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.
$$\Delta S=\int_a^b\frac{dQ}{T},\qquad (2)$$
Homework 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.
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?