# What is the origin of the concept of reduced mass?

I am looking for the origin of the concept of reduced mass as used in vibrational spectroscopy e.g. vibration of a diatomic molecule. Most of the texts simply define reduced mass as the sum of the inverses of masses of two bodies. Surely, this concept must have existed in mechanics long time ago. Does anyone know when and who introduced the concept of reduced mass? For the definition, see Reduced Mass

Thanks.

The use of reduced mass in spectroscopy goes back to Bohr's planetary model of the atom. Nasri explains the context in his notes on quantum mechanics:

"In 1912, Alfred Fowler showed that similar lines can be produced in a laboratory mixture of hydrogen and helium gas. Bohr noticed that they have the same spectrum of spectral lines as of hydrogen but with wavelength four times shorter and argued that they could correspond to the spectral lines of ionized helium. In Bohr’s theory this means that the corresponding Rydberg constant for the helium, $$\mathcal{R}_{He}$$, is four times $$\mathcal{R}_{H}$$ for the hydrogen atom. However, Fowler was not convinced and sent a letter to Nature where he pointed out that the ratio $$\mathcal{R}_{He}/\mathcal{R}_{H}$$ is not simply a factor of $$4$$, but instead is $$4.0016$$. Bohr understood that this small discrepancy is due to the fact that one neglected the effect of the finite mass of the nucleus. So, in the above expressions of $$W_n$$ and $$\nu_{rev}$$, the electron mass must be replaced by the reduced mass of the atom or the ion."

Bohr replied to Fowler also in a short note in Nature, The Spectra of Helium and Hydrogen (1913). The predicted value is $$4.00163$$, it is this agreement that allegedly prompted Einstein to say "This is a tremendous result. The theory of Bohr must then be right" (Pais, Inward Bound). The reduced mass expression is there, but not the name. However, many spectroscopic works in 1920-s already use "reduced mass" as a matter of course, some with reference to Bohr, e.g. Fine Structure of the Near Infra-Red Absorption Bands of the Halogen Acids by Colby (1920):

"Bohr has written the equilibrium conditions for such a system and for this problem it is only necessary to add one term to take care of the molecular rotation. This term is in fact of the same type as the one Bohr has used to indicate the centrifugal force on the electrons in the valence ring. $$2x$$ indicates the distance between the nuclei, $$y$$ the radius of the valence ring, $$M$$ is the reduced mass of the rotating system $$\left[I = M(2x)^2\right]$$, other symbols have their usual significance."

I could not pinpoint who coined the term "reduced mass", but it appears naturally when expressing the moment of inertia $$I$$ of two point masses. The moment of inertia is implicit in the work of Huygens, was introduced explicitly by Euler (1758), and later studied by Lagrange, Poinsot, Jacobi and others. Already Lagrange applied it to celestial mechanics (lunar libration, 1764). For a detailed account see J. L. Lagrange's Early Contributions to the Principles and Methods of Mechanics by Fraser:

"Although D'Alembert had used results about rigid bodies in his work in theoretical astronomy, the true foundations of the subject were laid by Euler in three important memoirs submitted to the Berlin Academy between 1750 and 1758. In the first of these memoirs Euler lays down the principle of linear momentum as his fundamental dynamical axiom and derives for a general rigid body the 'Euler' equations of motion relative to space fixed axes. In the later memoirs he introduces the concepts and terminology that have since become standard, the notions of principal axis, moment of inertia, Eulerian angle, etc. , and obtains the equations of motion with respect to reference axes fixed in the rigid body.

[...] In "Recherches sur la connoisance mecanique des corps" (1758) (op. tit. n. 43) Euler introduces the concept of 'moment of inertia' of a rigid body about a given line. He defines a 'principal axis' to be that line through the center of gravity for which the value of the moment of inertia is an extremum. This condition leads to two equations in the angles defining the position of the axis."

• Thanks, it is clear that the concept in mechanics existed long time ago. The question that remains is that who coined this term? – M. Farooq Nov 16 '19 at 20:44
• @M.Farooq Don't know. Google Scholar does not give any (relevant) hits on it before 1910s, so it could be that the concept was known and used without a label before Bohr's atom. I am not even sure if Bohr himself used it, or just his successors. – Conifold Nov 16 '19 at 20:50
• @M.Farooq I found Bohr's reply to Fowler, and added the reference. The reduced mass expression appears in his formula, but he does not call it that. – Conifold Nov 16 '19 at 21:10
• I think we should explore early rotational or vibrational spectroscopy papers too. The problem is that I don't know the pioneers of vibrational spectroscopy. Perhaps Herzberg's molecular spectra will give some clue. – M. Farooq Nov 16 '19 at 21:20
• @M.Farooq I found an earlier occurrence of the German version, reduzierte Masse, in a mechanics book by Thümmler, Fliehkraft und Beharrungsregler (1903). – Conifold Nov 16 '19 at 21:45