When we consider the various problems of integral calculus which naturally arise when we want to go deeper into the most different parts of physics, it is impossible not to be struck by the analogies that all these problems present to each other. Whether it concerns static or dynamic electricity, the propagation of heat, optics, elasticity, hydrodynamics, one is always led to differential equations of the same family and the boundary conditions, although different, are nevertheless not without offering some resemblances. We will cite only a few examples here.
I imagine first that one proposes to find the final temperature of a conductive, homogeneous and isotropic solid body, when the various points of the surface of this body are maintained artificially at given temperatures. This problem translates into analytic language if not this as follows:
Find a function V which in a portion of space satisfies the Laplace equation,
ΔV = d2V/dx2 + d2V/dy2 + d2V/dz2 = 0
and which takes given values at the various points of the surface which limits this space.
This is Dirichlet’s problem.
Suppose now that one seeks what is the distribution of static electricity on the surface of a given conductor; we shall encounter the same analytical problem.
It is a question of finding a function V which satisfies the Laplace equation in all the space exterior to the conductor and which reduces to 0 at infinity and to 1 at the surface to the conductor.
It is a special case of Dirichlet’s problem, but we know a way (by Green’s functions) to reduce the general case to this particular case.
The two problems, absolutely different from the physical point of view, are identical from the analytical point of view.
Other analogies, though less complete, are however obvious.
Source: H. 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. © 2022 Nicolae Sfetcu
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