A glimpse into the realm of Autonomous Harmonic Oscillators, constants of the motion and Fosse’s Aliss at the Fire
A couple of years ago, I was finishing Jon Fosse’s (2023 Nobel prize) “Aliss at the Fire”. Naturally, I could think of nothing else but harmonic oscillators and the undeniable, inherent beauty of periodicity.
Grief can trascend irreversibility. Signe’s mind broke the arrow of time. Her memory is caught in a cycle where pain precedes hope, and hope gives way to pain. This inescapable duality constitutes and irreducible reality, where longing dissolves the linearity of time: circle’s non-contractibility.
Asle’s disappearance is the alpha and omega of a universe reborn through its own iterative death: cosmic Dance of Shiva. That is “Aliss at the Fire”: repetition within resignation.
… and she, lying there on the bench, sees herself standing and looking at the flames in the stove and then she sees herself go over to the window and she sees herself stand there and look out the window and then she looks, standing there in front of the window, at the bedroom door and it opens… (Fosse, 2010)
I confess that, between Laplace’s Demon and Heisenberg’s uncertainty, I would choose determinism. Between dissipative forces and constants of motion, I am more inclined toward conservative systems. I dream of an ergodic, static, analytic ultimate reality -an oniric, homogeneous container where order is preserved through a symmetry yet to be unveiled.
Autonomous Harmonic Oscillators
Brief history
The origins of dynamics traces back to the study of mechanical systems. One of the most studied classical system is the harmonic oscillator. The study of oscillatory phenomena began with observations of natural, periodic systems like pendulums and vibrating strings. The Pythagoreans, who laid the groundwork for understanding harmonic motion in music, investigated the relationship between the pitch of a sound and the length of a vibrating string.
Around 1602, during the scientific revolution, Galileo noticed that, for small perturbations, the period of a pendulum’s swing is almost independent of its amplitude. That is how he designed a pendulum clock. In 1656 Huygens built the first pendulum clock. This engineer also derived the formula for the period of a simple pendulum while introducing the concept of isochronism and contributing to the study of synchronization phenomena (coupled oscillators). In 1678, Hooke’s work on springs (Hooke’s law) was published.
The mathematical formalization followed: Netwon’s “Principia” (laws of motion), Euler’s “De Novo Genere Oscillationum” (differential equations for mass-spring systems, wave propagation, strings , etc.), Bernoulli’s superposition principle for normal -harmonic- modes, Lagrange’s “Mécanique Analytique”, Hamilton, Fourier, etc.
Since then, the study of oscillators has been expanded to wave phenomena, practical applications and quantum mechanics. The simple harmonic oscillator became a fundamental model in modern physics (even atoms are modelled as oscillators). Oscillators are relevant nowadays: nonlinear and coupled oscillators revealing chaotic behavior, fields being treated as collections of oscillators in Quantum Field Theory and, in general, the harmonic oscillator playing a sine qua non role as a foundational physics topic (even to introduce or give an intuition regarding differential equations).
The model
For small displacements, a pendulum experiencing a frictional force is an example of a damped harmonic oscillator. A spring with a mass attached at one end and fixed at the other is an even simpler one.
Using Newtonian mechanics, we can model the damped harmonic oscillator as an ideal particle of mass m, moving, to and fro, along the x-axis (let’s keep it simple: 1-D), from its equilibrium point. Let x(t) be the position of the mass at time t and, let ẋ(t) and ẍ(t) be, respectively, the velocity and the acceleration of the mass.
By the second Newton’s law, the force acting on the mass is equal to the rate of change of its linear momentum: F = mẍ. In addition, the spring is particularly known by the restorative force, proportional to its displacement: move the spring from its equilibrium and it will “try” to “return” with a force according to Hooke’s Law: H = -kx, where k is a constant that tells how stiff the spring is.
Adding a resistive (dissipative) force R leads to a damped motion. This force, always opposite to velocity’s direction -since it slows down the mass-, can be thought as the friction that impedes the system to oscillate forever. Thus, R = -αẋ, with α, a real greater or equal to zero, being the damping constant.
Hence, we can express the equation of motion of our system as follows:
where Γ = α/(2m) is the damping parameter andω₀ = √{k/m} is the natural angular frequency of the oscillation, i.e.: the number of revolutions per second (inversely proportional to the period of the motion).
Notice that ẍ + 2Γẋ + ω₀² x = 0 is an homogeneous (autonomous or non-driven system), second order linear equation with constant coefficients and, therefore, with analytical solution. This equation can be easily solved by finding the transient state through the auxiliary quadratic equation λ² + 2Γλ + ω₀² = 0. If the discriminant ▵ > 0 (overdamped oscillator), then x(t) = Be^{λ₁t} + Ae^{λ₂t}. If ▵ = 0 (critical damping), then x(t) = (Bt+C)e^ {−ω₀t}. Lastly, if ▵ < 0 (underdamped oscillator), then x(t) = cos (α t + ϕ)Ae^{−2Γt}.
Now, if the dissipative force is zero (zero damping constant), then we get the equation of motion of a simple harmonic oscillator (purely imaginary solution):
And x(t) = A cos (ω₀ t )+ B sin (ω₀ t) is the position of the mass that will perpetually oscillate.
The beauty of the simple harmonic oscillator is evident: the phase space will show closed paths around a single point, indicating a quantity conserved, a constant of the motion, thus, endless cycles.
Constants of motion
Let’s formally define the concept “constant of the motion”:
If we are provided with K, we know that it is a constant of the motion if, for all x and y, thet total derivative of K is zero (time invariant):
The Simple Harmonic Oscillator is a conservative system in which energy is a conserved quantity. Let’s show this:
Now,
And, the total mechanical energy of our system is a constant of the motion.
Constants of motion are relevant when it comes to stability analysis of nonlinear systems in which fixed points are elliptic. By the Hartman-Grobman theorem, we know that, in the neighborhood of an hyperbolic fixed point, the behavior in the linearized system agrees with that of the nonlinear system. But, this is not true when it comes to centres. To find the nature of an elliptic point, it is useful to look for constants of the motion. If there is a constant of the motion, our equilibrium point is a centre.
It is your turn to show, through contants of the motion, that the total mechanical energy of a Damped Harmonic Oscillator is not time invariant.
References:
Dugas, R. (1988). A history of mechanics. Dover Publications.
Mach, E. (1919).The Science of Mechanics: A Critical and Historical Exposition of its Principles. (T. J. McCormack, Trans.). The Open Court Publishing Co
Taylor, J. R. (2005). Classical mechanics. University Science Books.
