Affine Geometry, Curve Flows, and Invariant Numerical Approximations

The present CriticalTBibliography,* which includes 2636 citations, is the twenty-fourth to be classified according to the system established in 1953. The main purpose of the classification has always been, in the words of its founder, George Sarton, "to satisfy the needs of historians of science in general rather than those of historians of particular sciences". While the classification is both chronological and by subject, preference is given to the former. The reader who wishes to find all references to a particular subject, therefore, must examine several sections of the bibliography.

One hundred third critical bibliography of the history of science and its cultural influences (to January 1978).

The Lie group invariance properties of the one‐dimensional radiation hydrodynamic equations with the equilibrium diffusion approximation, a local thermodynamical equilibrium assumption, and an arbitrary material equation of state are derived. These properties are used systematically to generate similarity solutions of these equations for a given form of the equation of state. A comprehensive list of allowed similarity solutions for a perfect gas is presented. Several special cases that have been found previously by other authors appear in the list. Many other cases not reported previously are also presented. An example numerical solution is given for a piston‐driven shock with a thermal precursor.

Lie group invariance properties of radiation hydrodynamics equations and their associated similarity solutions

It is a well known fact, documented in Lie’s correspondence with Adolph Mayer, that it was during the winter of 1873–74 that Lie began to systematically develop what became his theory of continuous transformation groups. Until recently, Lie’s decision to devote himself to this enterprise had struck me as rather mysterious. He had begun his career as a geometer and under the influence of Felix Klein had come to appreciate the potential importance that a theory of continuous transformation groups could have in applications to geometry and to the theory of differential equations.1 By 1871, however, very little of that potential had been realized. In [Klein, Lie 1870] Klein and Lie had called attention to a class of curves and surfaces defined by one and two parameter continuous groups of commuting projective transformations. These W-curves and W-surfaces (as they named them) included examples previously studied by other geometers. Klein and Lie suggested that the entire class was of special geometrical interest because of their invariance with respect to the group that generated them, but they apparently did not impress other geometers with their novel ideas, most of which were only sketched. They only published the details regarding W-curves in the plane [Klein, Lie 1871] because, in general, in order to determine the “W-configurations„ in n-dimensional projective space, they first had to classify the groups of commuting projective transformations that defined them, and already for n = 3 the classification problem became so involved that they abandoned their original plans to present a detailed discussion of W-curves and surfaces in space.

Jacobi and the birth of Lie's theory of groups

Using approximate symmetry methods for differential equations we have investigated the exact and approximate symmetries of a Lagrangian for the geodesic equations in the Kerr spacetime. Taking Minkowski spacetime as the exact case, it is shown that the symmetry algebra of the Lagrangian is 17 dimensional. This algebra is related to the 15 dimensional Lie algebra of conformal isometries of Minkowski spacetime. First introducing spin angular momentum per unit mass as a small parameter we consider first-order approximate symmetries of the Kerr metric as a first perturbation of the Schwarzschild metric. We then consider the second-order approximate symmetries of the Kerr metric as a second perturbation of the Minkowski metric. The approximate symmetries are recovered for these spacetimes and there are no non- trivial approximate symmetries. A rescaling of the arc length parameter for consistency of the trivial second-order approximate symmetries of the geodesic equations indicates that the energy in the charged-Kerr metric has to be rescaled and the rescaling factor is r-dependent. This re-scaling factor is compared with that for the Reissner–Nordstrom metric.

Approximate Noether symmetries of the geodesic equations for the charged-Kerr spacetime and rescaling of energy

Sophus Lie's 1880 transformation group paper

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Sophus Lie's 1880 transformation group paper