what is simple harmonic motion and its governing equation

What follows is grounded strictly in what the evidence states.

Rath et al. (2021)

• Classical Harmonic Oscillator — a system undergoing periodic motion whose phase portrait forms a closed loop, indicating stable oscillatory behaviour.

• Quantum-Mechanical Oscillator — the quantum analogue of the classical oscillator, whose monitoring is subject to the uncertainty principle (the "standard quantum limit") when using amplitude-and-phase measurement methods.

• Quantum-Classical Coupling — the interaction between quantum and classical degrees of freedom in a mixed oscillator system, studied via group-theoretical approaches and quantum-classical equations of motion.

• Nonlinear Damped Oscillator — a generalisation of SHM where external harmonic forces drive a multi-degree-of-freedom system, analysed via Lagrangian equations of motion and multiple-scales methodology. Abohamer et al. (2023)

Governing Equation of SHM

For a particle of mass $m$ subject to a linear restoring force $F = -kx$, Newton's second law gives:

$m\ddot{x} + kx = 0$

Dividing through by $m$:

$\ddot{x} + \omega^2 x = 0, \quad \omega = \sqrt{\frac{k}{m}}$

where $x$ is displacement, $\omega$ is the angular frequency, $k$ is the spring constant, and $m$ is the mass.

General Solution:

$x(t) = A\cos(\omega t + \phi)$

where $A$ is amplitude and $\phi$ is the initial phase.

Worked Numeric Example: Let $m = 1\ \text{kg}$, $k = 4\ \text{N/m}$.

$\omega = \sqrt{\frac{4}{1}} = 2\ \text{rad/s}, \quad T = \frac{2\pi}{\omega} = \pi \approx 3.14\ \text{s}$

If $A = 0.5\ \text{m}$ and $\phi = 0$, then $x(t) = 0.5\cos(2t)$.

• The classical harmonic oscillator produces a closed-loop phase portrait, confirming stable periodic motion.

• The quantum harmonic oscillator is subject to the standard quantum limit when monitored by amplitude-and-phase methods.

• Quantum-classical mixing of oscillators can be described via a bracket that preserves the Lie algebra structure of both quantum and classical mechanics.

• In nonlinear extensions, the equations of motion are derived using Lagrangian mechanics and solved approximately via multiple-scales methodology. Abohamer et al. (2023)