Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 5.2-1

THE EXTREMUM PRINCIPLE IN THE LEGENDRE TRANSFORMED REPRESENTATIONS

Physics — Statistical Mechanics · legendre-transform-parabola

Problem

The equation

y = x^2/10

describes a parabola.

a

) Find the equation of this parabola in the ``line geometry representation''

\psi = \psi(P)

b

) On a sheet of graph paper (covering the range roughly from

x \approx -15

x \approx +15

and from

y \approx -25

y \approx +25

) draw straight lines with slopes

P = 0

\pm 0.5

\pm 1

\pm 2

\pm 3

and with intercepts

\psi

satisfying the relationship

\psi = \psi(P)

as found in part (

a

). (Drawing each straight line is facilitated by calculating its intercepts on the

x

-axis and on the

y

-axis.)