Ordinary Differential Equations (17): Physics and Engineering Applications

Differential equations are not a pure mathematical game – they are the language for understanding the physical world. From celestial motion to circuit response, from a swinging pendulum to vortex shedding behind a bridge cable, every dynamical system “speaks” ODE.

This chapter is a deliberate tour through five canonical applications. Each one will pay back the entire ODE toolkit we built in chapters 1-16: phase planes, eigenvalues, Laplace transforms, modal analysis, conservation laws, numerical integration, control. None of the examples is a “toy” – they are all genuine working physics, written tightly so that the structure remains visible.

What You Will Learn

The nonlinear pendulum: small-angle linearisation vs the full equation, period vs amplitude, the separatrix
RLC circuits: the three damping regimes, resonance, the Q-factor, and tuning
Kepler orbits: gravity, eccentricity, conservation of energy and angular momentum, Bertrand’s theorem
Multi-DOF structural vibration: modal analysis, resonance, tuned mass dampers (the Tacoma lesson)
Fluid mechanics: Poiseuille flow, Reynolds number, vortex shedding, Stokes settling

Prerequisites

Chapters 1-3 – first- and second-order linear theory
Chapter 6-7 – linear systems and phase portraits
Chapter 8 – nonlinear stability and energy methods
Chapter 11 – numerical methods (used throughout)

1. The Nonlinear Pendulum – the Hello World of nonlinear ODEs

A point mass on a rigid rod of length $L$ in gravity $g$ obeys

$$ \ddot\theta + \frac{g}{L}\sin\theta = 0. $$

For small $\theta$ we replace $\sin\theta \approx \theta$ and recover simple harmonic motion with period $T_0 = 2\pi\sqrt{L/g}$ – independent of amplitude. That isochronism is the small-angle miracle Galileo exploited.

The full equation is not isochronous: the period depends on amplitude. The exact answer involves the complete elliptic integral of the first kind:

$$ T(\theta_0) = \frac{4}{\omega_0}\,K\!\bigl(\sin(\theta_0/2)\bigr), \qquad \omega_0 = \sqrt{g/L}. $$

Pendulum: large-angle vs linearised time histories, period vs amplitude, and the global phase portrait with libration / separatrix / rotation regions.

Top-left: from the same large initial angle ($\theta_0 = 1.5$ rad), the linear and full equations agree for one half-period and then drift apart – the linear model loses phase first, then amplitude. Top-right: $T(\theta_0)$ from numerical integration; below 0.5 rad the small-angle constant period is essentially perfect, but as $\theta_0 \to \pi$ the period diverges because the unstable equilibrium has been reached. Bottom: the full phase portrait $(\theta, \dot\theta)$. Inside the separatrix (red) the pendulum swings (libration); outside it goes over the top (rotation). The pattern repeats $2\pi$-periodically along $\theta$.

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from scipy.integrate import solve_ivp
import numpy as np

g, L = 9.81, 1.0

def pendulum(t, s):
    theta, omega = s
    return [omega, -(g/L)*np.sin(theta)]

# librations vs rotations: vary initial angular velocity at theta=0
for v0 in [1.5, 4.0, 6.5, 7.5]:    # last two go over the top
    sol = solve_ivp(pendulum, (0, 10), [0.0, v0],
                    t_eval=np.linspace(0, 10, 2000),
                    rtol=1e-10, atol=1e-12)
    # examine sol.y -- you will see closed loops or monotone theta

Why this matters. The pendulum is the simplest non-trivial nonlinear conservative system, and its phase portrait already contains every concept of Chapters 7-8: stable/unstable equilibria, separatrices, period-energy relations, integrability. Every other oscillator – molecular vibrations, Josephson junctions, plasma waves – borrows this skeleton.

2. RLC Circuits – the same equation, in copper

A series resistor-inductor-capacitor loop driven by voltage $V(t)$ obeys, for charge $q$ on the capacitor:

$$ L\ddot q + R\dot q + \frac{1}{C}\,q = V(t). $$

Compare with the mass-spring-damper $m\ddot x + c\dot x + kx = F(t)$. They are the same equation – so all our intuition about damping ratio and natural frequency carries over directly.

Define $\omega_n = 1/\sqrt{LC}$ (natural frequency) and $\zeta = (R/2)\sqrt{C/L}$ (damping ratio). The transfer function from $V$ to $q$ is

$$ H(s) = \frac{1/L}{s^2 + (R/L)\,s + 1/(LC)}, $$

with the same three step-response regimes from Chapter 16: underdamped, critical, overdamped.

Series RLC circuit: schematic, step responses, frequency response, and the Q-factor / bandwidth trade-off.

Top-left: the canonical series RLC. Top-right: step responses for three damping ratios; the underdamped case rings around the steady value, critical lands fastest without overshoot, overdamped is sluggish. Bottom-left: $|H(j\omega)|$ has a sharp resonant peak when $\zeta$ is small. Bottom-right: as $R$ rises, the Q factor $Q = (1/R)\sqrt{L/C}$ falls and the 3-dB bandwidth $\Delta\omega = R/L$ widens – a fundamental trade between selectivity and speed.

The peak amplitude of an undamped oscillator driven at resonance grows linearly in time – a phenomenon the Tacoma Narrows Bridge made historic in 1940 (more on that in section 4).

3. Planetary Orbits – where Newton meets Kepler

Newton’s gravitational law gives, in two dimensions,

$$ \ddot{\mathbf r} \;=\; -\frac{GM\,\mathbf r}{|\mathbf r|^3}. $$

The conserved quantities are total energy $E = \tfrac12|\mathbf v|^2 - GM/|\mathbf r|$ and angular momentum $\mathbf L = \mathbf r \times \mathbf v$. Together they prove Kepler’s three laws – without solving the ODE in closed form.

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from scipy.integrate import solve_ivp
import numpy as np

GM = 1.0
def kepler(t, s):
    x, y, vx, vy = s
    r = np.hypot(x, y)
    return [vx, vy, -GM*x/r**3, -GM*y/r**3]

# circle, ellipse, near-parabolic
for v0 in [1.0, 0.85, 0.55]:
    sol = solve_ivp(kepler, (0, 15), [1, 0, 0, v0],
                    t_eval=np.linspace(0, 15, 4000),
                    method='DOP853', rtol=1e-10)
    # sol.y[0:2] traces the orbit

Kepler orbits at different initial speeds; inverse-square vs inverse-cube collapse; numerical conservation of energy and angular momentum.

Left: starting at $(1,0)$ with varying tangential velocity, the orbit transitions from a circle to a thin ellipse – exactly the family Kepler observed. Top-right: a single conceptual experiment – replace $1/r^2$ with $1/r^3$, and the orbit no longer closes; it spirals into the centre (consistent with Bertrand’s theorem: only $1/r^2$ and the harmonic potential give closed orbits universally). Bottom-right: a high-order numerical integrator preserves both conserved quantities to roughly $10^{-7}$ over many revolutions.

Bertrand’s theorem is one of those quietly amazing facts: out of all spherically symmetric potentials, exactly two ($\propto 1/r$ and $\propto r^2$) produce closed orbits for every bound trajectory. That is why a stable Solar System with closed planetary loops exists. Mathematics first, astronomy second.

4. Structural Vibration – buildings, bridges, tuned mass dampers

A multi-storey building is, to first approximation, a chain of masses connected by stiff springs (the columns). For two storeys with masses $m_1, m_2$ and stiffnesses $k_1, k_2$:

$$ M\ddot{\mathbf x} + C\dot{\mathbf x} + K\mathbf x = \mathbf F(t), \qquad M = \begin{pmatrix} m_1 & 0 \\ 0 & m_2 \end{pmatrix}, \; K = \begin{pmatrix} k_1+k_2 & -k_2 \\ -k_2 & k_2 \end{pmatrix}. $$

Modal analysis. The eigenproblem $K\,\boldsymbol\phi = \omega^2 M\,\boldsymbol\phi$ yields natural frequencies $\omega_i$ and mode shapes $\boldsymbol\phi_i$. Decomposing $\mathbf x = \sum_i q_i(t)\,\boldsymbol\phi_i$ decouples the equations into $n$ independent SDOF oscillators – the multi-degree-of-freedom problem becomes $n$ scalar problems.

2-DOF building: mode shapes, free vibration, frequency response, resonant growth, and the effect of a tuned mass damper.

Left: the two mode shapes – mode 1 is in-phase (whole building sways), mode 2 is out-of-phase (storeys move opposite each other). Middle column: time response after kicking floor 2 (top), and frequency response with two clean resonance peaks (bottom). Right column: top – under resonant forcing without damping the amplitude grows linearly in time, the canonical Tacoma Narrows picture; bottom – adding a small tuned mass damper (TMD) tuned to the offending mode splits the single tall peak into two small adjacent peaks, the standard skyscraper-stabilisation trick (the 660-ton sphere atop Taipei 101).

The pattern is universal: whether the system is a violin string, a wing, a turbine blade or the Millennium Bridge, the steps are form $M, C, K$ -> solve eigenproblem -> decouple via modes -> add damping where needed.

5. Fluid Mechanics – from steady pipes to vortex shedding

Steady pipe flow (Poiseuille)

For incompressible viscous flow in a long circular pipe, the steady Navier-Stokes equations reduce to an ODE in the radial coordinate:

$$ \frac{1}{r}\frac{d}{dr}\!\Bigl(r\,\frac{du}{dr}\Bigr) = -\frac{1}{\mu}\frac{dp}{dz}, $$

with boundary conditions $u(R) = 0$ (no-slip) and $u(0)$ finite. The solution is the parabolic profile $u(r) = u_{\max}(1 - r^2/R^2)$, and total volume flow $Q = \pi R^4 \Delta p / (8\mu L)$ – the Hagen-Poiseuille law. The fourth-power dependence on radius explains why arteries dilate to drop blood pressure: doubling radius eightfold-improves flow at the same pressure gradient.

Reynolds number and the laminar-turbulent transition

The dimensionless number $\mathrm{Re} = \rho U L / \mu$ measures the ratio of inertial to viscous effects. Below $\mathrm{Re}\sim 2300$ flow is laminar and the linear $Q \propto \Delta p$ holds; above it the same pressure drop produces less flow because turbulent eddies dissipate energy.

Vortex shedding (the Strouhal number)

Behind a bluff body in steady flow, alternating vortices shed at frequency $f$ such that the Strouhal number $\mathrm{St} = f D / U$ is roughly constant ($\approx 0.21$) over a wide range of Re. When that frequency matches a structural mode, you get the Tacoma resonance again.

Settling sphere (Stokes drag)

A small dense sphere falling in viscous fluid satisfies

$$ m\dot v = (\rho_p - \rho_f) V g - 6\pi\mu R\,v, $$

a first-order linear ODE with terminal velocity $v_t = (\rho_p - \rho_f) V g / (6\pi\mu R)$ and time constant $\tau = m/(6\pi\mu R)$.

Top-left: parabolic velocity profile from Hagen-Poiseuille. Top-right: $Q$ vs $\Delta p$ shows the linearity of laminar flow breaking into a $\sqrt{\Delta p}$ scaling once turbulence dominates. Bottom-left: Strouhal number across a huge Reynolds-number range – the plateau near $0.21$ in the subcritical regime is the reason cables and chimneys oscillate at predictable frequencies. Bottom-right: spheres of three radii falling under gravity reach terminal velocity exponentially, with $v_t$ scaling as $R^2$.

6. The Common Thread

Every domain in this chapter speaks the same five-step grammar:

Write Newton’s / Kirchhoff’s / Navier-Stokes law for an infinitesimal element.
Reduce to ODEs by symmetry, dimensional analysis, or projection onto modes.
Identify the equilibrium and linearise; find natural frequencies and damping ratios.
Add forcing, look for resonance, and decide whether to add damping or de-tune the source.
Integrate numerically when the nonlinearity bites (large amplitude, turbulence, multi-body).

Master this loop and a vast portion of physics and engineering is unified.

Domain	Governing ODE	Key dimensionless number
Mechanics	$\ddot\theta + (g/L)\sin\theta = 0$	–
Electrical	$L\ddot q + R\dot q + q/C = V(t)$	$Q = (1/R)\sqrt{L/C}$
Orbital	$\ddot{\mathbf r} = -GM\mathbf r/	\mathbf r
Structural	$M\ddot x + C\dot x + Kx = F$	modal damping $\zeta_i$
Fluid (pipe)	$\nabla \cdot \boldsymbol\sigma = 0$ -> radial ODE	Re
Fluid (wake)	– (empirical)	St $\approx 0.21$

Summary

Differential equations are the lingua franca of physics. You spent 16 chapters building the language; this chapter showed how a handful of canonical models – pendulum, RLC, Kepler, MDOF building, pipe flow – carry you from undergraduate physics to professional engineering practice. None of them required new mathematics. They required us to recognise the structure and reach for the right tool.

The next chapter, the finale, looks beyond this classical menu: Neural ODEs, stochastic and fractional equations, and the questions that connect ODE theory to modern machine learning.

References

Kreyszig, Advanced Engineering Mathematics, Wiley (2011).
Taylor, Classical Mechanics, University Science Books (2005).
Goldstein, Poole & Safko, Classical Mechanics, Pearson (2002).
Den Hartog, Mechanical Vibrations, Dover (1985).
Acheson, Elementary Fluid Dynamics, Oxford (1990).

Previous Chapter: Chapter 16: Control Theory

Next Chapter: Chapter 18: Frontiers and Series Summary

This is Part 17 of the 18-part series on Ordinary Differential Equations.