The Legendre transform in geometric calculus Available to Purchase

This paper explores the extension of the Legendre transform from scalar calculus to geometric calculus. In physics, the Legendre transform provides a change of variables to express equations of motion or other physical relationships in terms of the most convenient dynamical quantities for a given experimental or theoretical analysis. In classical mechanics and in field theory, the Legendre transform generates the Hamiltonian function of a system from the Lagrangian function or vice versa. In thermodynamics, the Legendre transform allows thermodynamic relationships to be written in terms of alternative sets of independent variables. In this paper, we review the properties of the Legendre transform in scalar calculus and show how an analogous transformation with similar properties may be constructed in geometric calculus.