Mention of Bernoulli principle by Bill Lawvere

Question

In the Author Commentary to the reprint of the paper paper Diagonal Arguments and Cartesian Closed Categories in Theory and Applications of Categories Bill Lawvere wrote:

Although the cartesian-closed view of function spaces and functionals was intuitively obvious in all but name to Volterra and Hurewicz (and implicitly to Bernoulli), it has counterexamples within the rigid framework advocated by Dieudonné and others. According to that framework the only acceptable fundamental structure for expressing the cohesiveness of space is a contravariant algebra of open sets or possibly of functions. Even though such algebras are of course extremely important invariants, their nature is better seen as a consequence of the covariant geometry of figures. Specific cases of this determining role of figures were obvious in the work of Kan and in the popularizations of Hurewicz’s k-spaces by Kelley, Brown, Spanier, and Steenrod, but in the present paper I made this role a matter of principle: the Yoneda embedding was shown to preserve cartesian closure, and naturality of functionals was shown to be equivalent to Bernoulli’s principle.

First of all, as if I understand correctly the mention of Bernoulli refers to Johann Bernoulli (1667 – 1748) which is known for his contributions to infinitesimal calculus and educating Leonhard Euler in the pupil's youth. Am I correct thinking so?

Secondly, which principle exactly does the author mention in the passage? The only Bernulli principle which comes on my mind is Daniel Bernoulli's principle in fluid dynamics.

Answer 1

Since the topic of Lawvere paper is differential geometry, the Bernoulli is likely Jacob Bernoulli, or his brother Johann, and refers to their calculus of variations and the principle of virtual work. Further evidence is this paper by Lawvere on category theory, where he explicitly refers to Bernoulli in the context of the calculus of variations.

Answer 2

Extended remarks in support of Carlos' answer:

From "The Legendre-Fenchel transform from a category theoretic perspective" by Simon Willerton (pg. 2),

... Lawvere [10] showed using enriched category theory that category theory was entwined with metric space theory.

Following Lawvere [11] and taking enriched categories seriously, we will see here how, from the pairing between a vector space and its dual, the nucleus of a profunctor construction [17] gives rise to a large amount of the theory around the Legendre-Fenchel transform and convex functions.

The Legendre-Fenchel transform, calculus of variations, geodesics, principles of virtual work attributed to d'Alembert and Johann Bernoulli, Lagranges's principle of least action, and the dual formulations of Lagrange's and Newton's equations of motion and the associated Hamiltonian total energy (H=T+V) and Lagrangian action (L=T-V) related via the Legendre-Fenchel transform are all intimately intertwined.

Does anyone have additional refs on these connections with category theory?