# Mathematicians on the geometrization of the Hamilton–Jacobi formalism

# Mathematicians on the geometrization of the Hamilton–Jacobi formalism

It has long since been remarked by mathematicians that William Rowan Hamilton's method contains purely geometrical truths, and that a peculiar mode of expression, suitable to it, is required in order to express these clearly. But this fact has only come to light in a somewhat perplexing form, namely, in the analogies between ordinary mechanics and the geometry of space of many dimensions. Together with an explicit reference to the work of Eugenio Beltrami, Rudolf Lipschitz, and Jean-Gaston Darboux, Heinrich Hertz made only one other reference to the work on mechanics done by contemporary mathematicians. Hertz correctly connected their work with his own treatment of the Hamiltonian formalism. This chapter gives a short summary of the mathematical developments in Hamilton-Jacobi formalism during the period 1828-1888 and compares them with those of Hertz. The views of Johann Carl Friedrich Gauss and Hamilton on geodesics, optics, and dynamics are discussed, along with those of Joseph Liouville and Lipschitz on the principle of least action, and trajectories as geodesics.

*Keywords:*
William Rowan Hamilton, Hamilton-Jacobi formalism, mechanics, geometry, mathematicians, Johann Carl Friedrich Gauss, geodesics, optics, principle of least action

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