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ALS in German, signed “A. Einstein,” one page, 8.75 x 11, undated. Handwritten letter to mathematician Chaim Herman Müntz, densely penned with equations and mathematical notation as Einstein works through a problem involving Hamiltonian mechanics and invariance, in full (translated): "The equation proposed by me is, after all, largely covariant. Namely, the canonical equations are covariant: dqᵢ/dt = ∂H/∂pᵢ, dpᵢ/dt = −∂H/∂qᵢ
If one restricts oneself to transformations for which Σᵢ pᵢ dqᵢ is invariant, then because of the first system of canonical equations, Σ pᵢ ∂H/∂pᵢ is also [“covariant” crossed out] invariant, because ∂H/∂pᵢ transforms like dqᵢ. Regarding the left side of the equation, it behaves as follows: The equation ∂(ρ dqᵢ/dt)/∂qᵢ + ∂(ρ dpᵢ/dt)/∂pᵢ = ∂ρ/∂t is an invariant equation because of the invariant sense of this equation, specifically for arbitrary canonical transformations, and therefore also the equation ∂ρ/∂qᵢ ∂H/∂pᵢ − ∂ρ/∂pᵢ ∂H/∂qᵢ = ∂ρ/∂t.
The left side of this equation thus transforms like ∂ρ/∂t, and therefore also like ρ. Thus, 1/ρ (∂ρ/∂qᵢ ∂H/∂pᵢ − ∂ρ/∂pᵢ ∂H/∂qᵢ) is an invariant with respect to arbitrary transformations. It follows, therefore, that the equation ∂ρ/∂qᵢ ∂H/∂pᵢ − ∂ρ/∂pᵢ ∂H/∂qᵢ = − j 2π/h pᵢ ∂H/∂pᵢρ is invariant for all transformations which leave pᵢ dqᵢ invariant. We probably cannot demand more than that. It is therefore likely justified if we attempt the problem of the rotator on the basis of this equation." In fine condition. Accompanied by the original mailing envelope postmarked in Potsdam, Germany, apparently in 1929.
Einstein had been corresponding with mathematician Chaim Herman Müntz since 1927, and by 1928 the two were exchanging ideas concerning distant parallelism, or Fernparallelismus, the mathematical framework Einstein was then exploring in his attempt to formulate a unified theory of gravitation and electromagnetism. Müntz contributed directly to these investigations, undertaking detailed calculations of the centrally symmetric problem that Einstein acknowledged in his published work. Their collaboration continued during an especially active period in Einstein’s search for a unified field theory, when he pursued the distant-parallelism approach in a series of papers between 1928 and 1931. Later in 1929, Hamilton’s principle became another element of Einstein’s unified-field research as he addressed mathematical concerns raised by Müntz and Cornelius Lanczos.
The mathematics developed in the present letter, however, appears to address a distinct problem rather than Einstein’s unified-field theory itself. Einstein begins with Hamilton’s canonical equations of classical mechanics and examines their behavior under transformations preserving certain expressions. He proceeds through a succession of covariance and invariance arguments, determining how individual terms transform and establishing the invariance of increasingly complex expressions before arriving at his final equation. The page consequently preserves the progression of Einstein’s mathematical reasoning almost step by step, from the canonical equations through the transformations necessary to establish his result. He concludes, “We probably cannot demand more than that,” before proposing that they attempt what he calls the problem of the rotator on the basis of the equation he has derived.
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