This monograph studies the idea that there is only a single concept of mass in physics. Notably, this is so not since Albert Einstein, as is the canonical reading, but since Isaac Newton. Sir Isaac Newton in his Principia (1687) introduced mass as a measure of inertia.

Newton then deduced that mass also measures weight.

Newton also proved that this very same inertial mass finds itself at the source of gravitational attraction. Hence, the twentieth-century distinction between inertial, gravitational and gravitating mass may be didactically useful, but does no justice to the legacy of Isaac Newton. The monograph discusses two aspects of why later readers missed the true content of Newton's legacy. Firstly, because of Newton's redactional choices. Secondly, because of the historical and socio-scientific context: the expectations and attitudes of Newton's peers, readers and criticasters. This is done in the context of the history of Western thought, i.e. philosophy. ... Read more

Legendre’s equation is key in various branches of physics. Its general solution is a linear function space, spanned by the Legendre functions of the first and second kind. In physics, however, commonly the only acceptable members of this set are Legendre polynomials. The quantization of the eigenvalues of Legendre’s operator is a consequence of this. We present and explain a stand-alone and in-depth argument for rejecting all solutions of Legendre’s equation in physics apart from the polynomial ones. We show that the combination of the linearity, the mirror symmetry and the signature of the regular singular points of Legendre’s equation are quintessential to the argument. We demonstrate that the evenness or oddness of Legendre polynomials is a consequence of the same premises. ... Read more

We enhance Frobenius’ method for solving linear ordinary differential equations about regular singular points. Key to Frobenius’ approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel’s equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order. ... Read more

We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly,
why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory.
In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rank-free, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion
for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a
continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless. ... Read more