Calculus of variations In his doctoral dissertation, Carathéodory showed how to extend solutions to discontinuous cases and studied isoperimetric problems. Carathéodory constructed his method for deriving sufficient conditions based on the use of the Hamilton–Jacobi equation to construct a field of extremals. The ideas are closely related to light propagation in optics. The method became known as ''Carathéodory's method of equivalent variational problems
or the royal road to the calculus of variations''. A key advantage of Carathéodory's work on this topic is that it illuminates the relation between the calculus of variations and partial differential equations.
Convex geometry for a square in
R2.
Carathéodory's theorem in convex geometry states that if a point x of \mathbb{R}^d lies in the
convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset P' of P consisting of d + 1 or fewer points such that x lies in the convex hull of P'. Equivalently, x lies in an r-
simplex with vertices in P, where r \leq d. The smallest r that makes the last statement valid for each x in the convex hull of
P is defined as the ''Carathéodory's number'' of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained. He is credited with the authorship of the
Carathéodory conjecture claiming that a closed convex surface admits at least two
umbilic points. The conjecture was proven in 2024 by Brendan Guilfoyle and Wilhelm Klingenberg.
Real analysis He proved an
existence theorem for the solution to ordinary differential equations under mild regularity conditions. Another theorem of his on the derivative of a function at a point could be used to prove the
Chain Rule and the formula for the
derivative of inverse functions.
Complex analysis He greatly extended the theory of
conformal transformation proving his
theorem about the extension of conformal mapping to the boundary of Jordan domains. In studying boundary correspondence he originated the theory of
prime ends. In 1909, he published a pioneering work "Investigations on the Foundations of Thermodynamics" in which he formulated the second law of thermodynamics axiomatically, that is, without the use of Carnot engines and refrigerators and only by mathematical reasoning. This is yet another version of the second law, alongside the statements of
Clausius, and of
Kelvin and Planck. Carathéodory's version attracted the attention of some of the top physicists of the time, including Max Planck, Max Born, and Arnold Sommerfeld. Max Born acclaimed this "first axiomatically rigid foundation of thermodynamics" and he expressed his enthusiasm in his letters to Einstein. in that while he was impressed by Carathéodory's mathematical prowess, he did not accept that this was a fundamental formulation, given the statistical nature of the second law. He formulated the axiomatic principle of irreversibility in thermodynamics stating that inaccessibility of states is related to the existence of entropy, where temperature is the integration function. The
second law of thermodynamics was expressed via the following axiom: "In the neighbourhood of any initial state, there are states which cannot be approached arbitrarily close through adiabatic changes of state." In this connexion he coined the term
adiabatic accessibility.
Optics Carathéodory's work in
optics is closely related to his method in the calculus of variations. In 1926 he gave a strict and general proof that no system of lenses and mirrors can avoid
aberration, except for the trivial case of plane mirrors. In his later work he gave the theory of the
Schmidt telescope. In his
Geometrische Optik (1937), Carathéodory demonstrated the equivalence of Huygens' principle and Fermat's principle starting from the former using Cauchy's theory of characteristics. He argued that an important advantage of his approach was that it covers the integral invariants of
Henri Poincaré and
Élie Cartan and completes the
Malus law. He explained that in his investigations in optics,
Pierre de Fermat conceived a minimum principle similar to that enunciated by
Hero of Alexandria to study reflection.
Historical During the Second World War Carathéodory edited two volumes of
Euler's Complete Works dealing with the Calculus of Variations which were submitted for publication in 1946. == The University of Smyrna ==