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In the book "The Road to Reality", Penrose claims that Joseph Lagrange was aware of Hamiltonian mechanics. Is there any evidence that this statement is true?

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Yes, he was, at least in a particular example. Near the end of his life, in 1808-09, Lagrange studied perturbative dynamics of a planet on an elliptic orbit, and derived what came to be called Hamiltonian equations for it in Second mémoire sur la théorie de la variation des constantes arbitraires dans les problèmes de mécanique, dans lequel on simplifie l'application des formules générales à ces problèmes (1809). A modern discussion with details can be found in Weinstein's Symplectic Geometry:

"The first symplectic manifold was introduced by Lagrange [LAI] in 1808. In studying the motion of the planets under the influence of their mutual gravitational interaction, he took as a starting point the elliptical motion of a single planet around the sun. (Actually, the focus of the ellipse is the center of the mass of the sun-planet system, but we will ignore this point here.) This ellipse was then considered to "drift" under the disturbing influence of the other planets. The drift was described by a system of differential equations, and Lagrange sought to put these equations in the simplest possible form.

Lagrange described the motion by six elements (real functions) $a_i$, introduced what is now called Lagrange brackets for them, and was able to write the equations in the form $\frac{da_i}{dt}=\sum_j b_{ij}\frac{\partial\Phi}{\partial a_j}$, where $b_{ij}$ was antisymmetric. "At the time of Lagrange, this advance was of advantage largely because it shortened by a factor of six some rather complicated computations". The $\Phi$ was what Lagrange "called the disturbing function, which depends on the perturbing forces but is independent of the choice of elements". The equations meant that the total time derivative of $\Phi$ along the trajectories was $0$, i.e. it was conserved. In 1809 he found a change of variables from $a_i$ to $q_i$, $p_i$, that satisfied the canonical commutation relations for his brackets. In terms of these elements, the equations he derived "are known today as Hamilton's equations", with $\Phi$ as the Hamiltonian.

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  • $\begingroup$ Do you know if Lagrange brackets were discovered before Poisson brackets ? $\endgroup$
    – GEP
    Commented Jan 22, 2020 at 18:14
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    $\begingroup$ @GEP Poisson introduced his brackets just a year later, in Sur la variation des constantes arbitraires dans les questions de mécanique (1809). Both were stimulated by each other's work in 1808-10, see Marle, The inception of symplectic geometry. $\endgroup$
    – Conifold
    Commented Jan 22, 2020 at 20:39
  • $\begingroup$ So Lagrange introduced his brackets first? Were there any direct predecessors to the brackets concept? Thanks for answering my previous questions, I highly appreciate it. $\endgroup$
    – GEP
    Commented Jan 22, 2020 at 21:43
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    $\begingroup$ @GEP In 1808. Marle gives some pre-history, not sure what would count as "predecessors". $\endgroup$
    – Conifold
    Commented Jan 22, 2020 at 21:57

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