The theory of elasticity

The theory of elasticity offers us more complicated equations, but which do not differ much from the preceding ones.

We still have three unknown functions u, v, w, to which I will add the auxiliary function Θ = du/dx + dv/dy + dw/dz and three equations, the first of which is written:

Δu + λ dΘ/dx = 0

I do not write the boundary conditions quite analogous, mutatis mutandis, to those of the previous problems and I pass immediately to a very important question in hydrodynamics and which consists in finding the components of the velocity at all points of a liquid when we know the components of the vortex at all the points of this same liquid. This problem from the analytical point of view is stated as follows:

Knowing three functions ɑ, β, γ find three unknown functions u, v, w, which satisfy certain boundary conditions and furthermore the equations

(1)   ɑ = dw/dy – dv/dz , β = du/dz – dw/dx , γ = dv/dx – du/dy , du/dx + dv/dy + dw/dz = 0

The analogy with the previous problems does not appear, it becomes so if we observe that the first three equations (1) can (by virtue of the fourth) be replaced by three others, the first of which is written

Δu = dβ/dz – dγ/dy

and of which others can be written by symmetry.

I could also show, if I were not afraid to tire the attention by too many examples, that almost all the questions, still poorly studied, relating to electrodynamic induction in nonlinear conductors come down to problems analogous to the preceding ones, and especially to the last question of hydrodynamics that I have just mentioned.

Source: Henri Poincaré, Sur les Equations aux Dérivées Partielles de la Physique Mathématique, American Journal of Mathematics, Mar., 1890, Vol. 12, No. 3 (Mar., 1890), pp. 211-294. Translation by Nicolae Sfetcu. © 2023 Nicolae Sfetcu