Harmonic oscillators are one of the fundamental elements of physical theory. They arise so often in so many different contexts that they can be viewed as a central paradigm that spans all aspects of physics. Some famous physicists have been quoted to say that the entire universe is composed of simple harmonic oscillators (SHO).
Despite the physicist’s love affair with it, the SHO is pathological! First, it has infinite frequency degeneracy which makes it prone to the slightest perturbation that can tip it into chaos, in contrast to non-harmonic cyclic dynamics that actually protects us from the chaos of the cosmos (see my Blog on Chaos in the Solar System). Second, the SHO is nowhere to be found in the classical world. Linear oscillators are purely harmonic, with a frequency that is independent of amplitude—but no such thing exists! All oscillators must be limited, or they could take on infinite amplitude and infinite speed, which is nonsense. Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light. Relativistic effects would modify the linearity, especially through time dilation effects, rendering the harmonic oscillator anharmonic.
Despite the physicist’s love affair with it, the SHO is pathological!
Therefore, for students of physics as well as practitioners, it is important to break the shackles imposed by the SHO and embrace the anharmonic harmonic oscillator as the foundation of physics. Here is a brief survey of several famous anharmonic oscillators in the history of physics, followed by the mathematical analysis of the relativistic anharmonic linear-spring oscillator.
Anharmonic oscillators have a long venerable history with many varieties. Many of these have become central models in systems as varied as neural networks, synchronization, grandfather clocks, mechanical vibrations, business cycles, ecosystem populations and more.
Already by the mid 1600’s Christiaan Huygens (1629 – 1695) knew that the pendulum becomes slower when it has larger amplitudes. The pendulum was one of the best candidates for constructing an accurate clock needed for astronomical observations and for the determination of longitude at sea. Galileo (1564 – 1642) had devised the plans for a rudimentary pendulum clock that his son attempted to construct, but the first practical pendulum clock was invented and patented by Huygens in 1657. However, Huygens’ modified verge escapement required his pendulum to swing with large amplitudes, which brought it into the regime of anharmonicity. The equations of the simple pendulum are truly simple, but the presence of the sinθ makes it the simplest anharmonic oscillator.
Therefore, Huygens searched for the mathematical form of a tautochrone curve for the pendulum (a curve that is traversed with equal times independently of amplitude) and in the process he invented the involutes and evolutes of a curve—precursors of the calculus. The answer to the tautochrone question is a cycloid (see my Blog on Huygen’s Tautochrone Curve).
Hermann von Helmholtz
Hermann von Helmholtz (1821 – 1894) was possibly the greatest German physicist of his generation—an Einstein before Einstein—although he began as a medical doctor. His study of muscle metabolism, drawing on the early thermodynamic work of Carnot, Clapeyron and Joule, led him to explore and to express the conservation of energy in its clearest form. Because he postulated that all forms of physical processes—electricity, magnetism, heat, light and mechanics—contributed to the interconversion of energy, he sought to explore them all, bringing his research into the mainstream of physics. His laboratory in Berlin became world famous, attracting to his laboratory the early American physicists Henry Rowland (founder and first president of the American Physical Society) and Albert Michelson (first American Nobel prize winner).
Even the simplest of simple harmonic oscillators would be limited by nothing other than the speed of light.
Helmholtz also pursued a deep interest in the physics of sensory perception such as sound. This research led to his invention of the Helmholtz oscillator which is a highly anharmonic relaxation oscillator in which a tuning fork was placed near an electromagnet that was powered by a mercury switch attached to the fork. As the tuning fork vibrated, the mercury came in and out of contact with it, turning on and off the magnet, which fed back on the tuning fork, and so on, enabling the device, once started, to continue oscillating without interruption. This device is called a tuning-fork resonator, and it became the first door-bell buzzers. (These are not to be confused with Helmholtz resonances that are formed when blowing across the open neck of a beer bottle.)
Baron John Strutt, the Lord Rayleigh (1842 – 1919) like Helmholtz also was a generalist and had a strong interest in the physics of sound. He was inspired by Helmholtz’ oscillator to consider general nonlinear anharmonic oscillators mathematically. He was led to consider the effects of anharmonic terms added to the harmonic oscillator equation. in a paper published in the Philosophical Magazine issue of 1883 with the title On Maintained Vibrations, he introduced an equation to describe the self-oscillation by adding an extra term to a simple harmonic oscillator. The extra term depended on the cube of the velocity, representing a balance between the gain of energy from a steady force and natural dissipation by friction. Rayleigh suggested that this equation applied to a wide range of self-oscillating systems, such as violin strings, clarinet reeds, finger glasses, flutes, organ pipes, among others (see my Blog on Rayleigh’s Harp.)
The first systematic study of quadratic and cubic deviations from the harmonic potential was performed by the German engineer George Duffing (1861 – 1944) under the conditions of a harmonic drive. The Duffing equation incorporates inertia, damping, the linear spring and nonlinear deviations.
Duffing confirmed his theoretical predictions with careful experiments and established the lowest-order corrections to ideal masses on springs. His work was rediscovered in the 1960’s after Lorenz helped launch numerical chaos studies. Duffing’s driven potential becomes especially interesting when α is negative and β is positive, creating a double-well potential. The driven double-well is a classic chaotic system (see my blog on Duffing’s Oscillator).
Balthasar van der Pol
Autonomous oscillators are one of the building blocks of complex systems, providing the fundamental elements for biological oscillators, neural networks, business cycles, population dynamics, viral epidemics, and even the rings of Saturn. The most famous autonomous oscillator (after the pendulum clock) is named for a Dutch physicist, Balthasar van der Pol (1889 – 1959), who discovered the laws that govern how electrons oscillate in vacuum tubes, but the dynamical system that he developed has expanded to become the new paradigm of cyclic dynamical systems to replace the SHO (see my Blog on GrandFather Clocks.)
Turning from this general survey, let’s find out what happens when special relativity is added to the simplest SHO .
Relativistic Linear-Spring Oscillator
The theory of the relativistic one-dimensional linear-spring oscillator starts from a relativistic Lagrangian of a free particle (with no potential) yielding the generalized relativistic momentum
The Lagrangian that accomplishes this is 
where the invariant 4-velocity is
When the particle is in a potential, the Lagrangian becomes
The action integral that is minimized is
and the Lagrangian for integration of the action integral over proper time is
The relativistic modification in the potential energy term of the Lagrangian is not in the spring constant, but rather is purely a time dilation effect. This is captured by the relativistic Lagrangian
where the dot is with respect to proper time τ. The classical potential energy term in the Lagrangian is multiplied by the relativistic factor γ, which is position dependent because of the non-constant speed of the oscillator mass. The Euler-Lagrange equations are
where the subscripts in the variables are a = 0, 1 for the time and space dimensions, respectively. The derivative of the time component of the 4-vector is
From the derivative of the Lagrangian with respect to speed, the following result is derived
where E is the constant total relativistic energy. Therefore,
which provides an expression for the derivative of the coordinate time with respect to the proper time where
The position-dependent γ(x) factor is then
The Euler-Lagrange equation with a = 1 is
providing the flow equations for the (an)harmonic oscillator with respect to proper time
This flow represents a harmonic oscillator modified by the γ(x) factor, due to time dilation, multiplying the spring force term. Therefore, at relativistic speeds, the oscillator is no longer harmonic even though the spring constant remains truly a constant. The term in parentheses effectively softens the spring for larger displacement, and hence the frequency of oscillation becomes smaller.
The state-space diagram of the anharmonic oscillator is shown in Fig. 3 with respect to proper time (the time read on a clock co-moving with the oscillator mass). At low energy, the oscillator is harmonic with a natural period of the SHO. As the maximum speed exceeds β = 0.8, the period becomes longer and the trajectory less sinusoidal. The position and speed for β = 0.9999 is shown in Fig. 4. The mass travels near the speed of light as it passes the origin, producing significant time dilation at that instant. The average time dilation through a single cycle is about a factor of three, despite the large instantaneous γ = 70 when the mass passes the origin.
 W. Moreau, R. Easther, and R. Neutze, “RELATIVISTIC (AN)HARMONIC OSCILLATOR,” American Journal of Physics, Article vol. 62, no. 6, pp. 531-535, Jun (1994)
 D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd. ed. (Oxford University Press, 2019)