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Coupled oscillators can be broken down into a superposition of normal mode oscillations. Who was the first person to solve for this system in this way?

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  • $\begingroup$ It is usually attributed to Joseph Louis Lagrange. $\endgroup$ – Alexandre Eremenko Dec 11 '16 at 3:46
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Edited. The story is long and complicated. From the mathematical point of view, three statements are involved:

A. Eigenvalues of a self-adjoint operator are real and simple,

B. There exists an orthogonal basis consisting of eigenvectors,

C. Every quadratic form can be reduced to principal axes.

In 1715 Brook Taylor found that functions $$\sin\frac{\pi nx}{a}\cos\frac{\pi nct}{a}$$ describe the vibrations of a homogeneous string of length $a$ (normal modes).

In 1750 Daniel Bernoulli proposed that the general solution of the equation of a homogeneous string is a linear combination (possibly infinite) of these normal modes. He used the previous work of d'Alembert on solution of linear ODE with constant coefficients. This conjecture of Bernoulli triggered a long discussion between Euler, d'Alembert, Bernoulli on representability of an arbitrary function by trigonometric series and on the general notion of function.

In 1759-61 Joseph Louis Lagrange proved Bernoulli's conjecture by considering an approximation of the string by a thread with beads of equidistant equal masses, solving the corresponding linear ODE explicitly, and passing to the limit.

In 1748 Leonard Euler showed that an arbitrary quadratic form in dimension 3 can be reduced to principal axes.

In 1787 Pierre Simon Laplace stated that all eigenvalues of a symmetric matrix are real, but his proof involved physical arguments.

In 1829 Augustin-Louis Cauchy proved that all eigenvalues of a symmetric matrix are real, and simple. He writes that Charles Sturm has a simpler proof to be published soon. I was unable to locate this proof. Cauchy is aware of the significance of this theorem for mechanics and for reduction of quadratic forms.

In 1813 Simeon Denis Poisson found the modern proof of reality of eigenvalues for self-adjoint operators. (He spoke of differential operators, but his proof applies to the general case. This is the proof given in modern linear algebra and functional analysis books).

In 1836 Charles Sturm proved A and B for ordinary linear differential operators of second order.

One had to wait till 20th century for a complete proof of B (the Spectral Theorem) in full generality in infinite dimensional space. The final form is due to von Neumann.

References. I was unable to find a precise reference on B. Taylor, d'Alembert, D. Bernoulli and on the work of Sturm which Cauchy mentions. Description of the work of Lagrange with references to his Oeuvres is here:

https://arxiv.org/pdf/1305.3438.pdf

The work of Euler, Laplace and Cauchy is described in the book Emerging mathematics, by J. Stedall. The reference on Cauchy is:

Sur l'equation a l'aide de laquelle on determine les inegalites seculaires de movment des planetes, Exercises de mathematiques 1829, 140-142, 159-160.

A very good reference for the whole story is

Dieudonne, History of functional analysis, North Holland 1981.

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