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)

The Nonlinear Pendulum — the Hello World of nonlinear ODEs#

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.

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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)]

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.

RLC Circuits — the same equation, in copper#

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.

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

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).

Planetary Orbits — where Newton meets Kepler#

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]

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

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.

Structural Vibration — buildings, bridges, tuned mass dampers#

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.

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.

Fluid Mechanics — from steady pipes to vortex shedding#

Steady pipe flow (Poiseuille)#

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

The Common Thread#

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

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

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

Worked Example with Units: Designing a Tuned RLC Bandpass#

A symbolic equation is mathematics; an equation with units attached is engineering. Design a series RLC bandpass with centre frequency $f_0 = 1\,\text{kHz}$ and bandwidth $\Delta f = 100\,\text{Hz}$ .

For $L\ddot q + R\dot q + q/C = V(t)$ , $\omega_0 = 1/\sqrt{LC}$ and $Q = \omega_0 L/R$ .

Step 3: check units. $\omega_0^2 L C = (\text{s}^{-2})(\text{H})(\text{F}) = (\text{s}^{-2})(\text{V}\cdot\text{s/A})(\text{A}\cdot\text{s/V}) = 1$ . Dimensionless, correct.

Step 4: physical reading. $Q = 10$ means after the source disconnects, oscillation amplitude takes about $Q/\pi \approx 3.2$ periods to fall by $1/e$ — about 3.2 ms of ringing. In math language, damping ratio $\zeta = 1/(2Q) = 0.05$ , deep underdamped.

Step 5: energy view. At resonance, peak inductor energy $\tfrac12 L I_{\max}^2$ swaps periodically with peak capacitor energy $\tfrac12 C V_{\max}^2$ , with $\pi/Q$ of it lost to $R$ each cycle. Higher $Q$ means narrower, sharper peak.

Five steps and four numbers turn an abstract second-order ODE into something I can put on a breadboard.

Limit Cases: Small Parameters Make Equations Degenerate#

The nonlinear pendulum $\ddot\theta + (g/\ell)\sin\theta = 0$ degenerates to the harmonic oscillator as $|\theta| \to 0$ , with amplitude-independent period $2\pi\sqrt{\ell/g}$ . Standard small-angle story.

What is more useful in the ML era is the other direction: parameters going to 0 or infinity often make the equation worse for numerical solvers, not better.

Case 1: high Reynolds number. As viscosity $\nu \to 0$ in Navier-Stokes, the equation formally reduces to Euler, but in practice almost all smooth solutions destabilise into turbulence. Numerically, small $\nu$ blows up the grid Péclet number; you must switch to upwind schemes or shock capturing.

Case 2: stiff spring limit. Spring constant $k \to \infty$ effectively constrains the mass to a manifold. Direct integration demands $h \sim 1/\sqrt k$ — unaffordable. The physical fix: rewrite as a constrained ODE/DAE with Lagrange multipliers. scipy.integrate.solve_ivp with LSODA or BDF survives moderate stiffness; beyond that you need Hairer-Wanner-style implicit symplectic methods.

Case 3: low Mach number. Compressible flow at $M \to 0$ has acoustic and convective timescales separated by $1/M$ . Explicit CFL is dominated by sound waves and wastes 99% of the work. Low-Mach preconditioning is a core trick of industrial CFD.

Lesson: a parameter going to zero rarely simplifies the numerics. This is the contradiction Neural Operators in PDE-ML chapter 2 are trying to dodge.

Data-Driven Modelling: SINDy and Koopman#

Section 6 made the case that one ODE skeleton describes five phenomena. In real engineering, when a new system shows up the equation may not be derivable. You go from data to equation.

Most components of $\xi$ collapse to zero; the survivors give you the equation form. With pysindy, 1000 samples of Lorenz data recover the three RHS terms with their coefficients $\sigma, \rho, \beta$ — provided the library includes the correct basis.

Koopman operator methods. Lift a nonlinear map $x_{t+1} = F(x_t)$ to a linear operator $\mathcal{K}\,g = g \circ F$ on observable functions. Combined with DMD (Dynamic Mode Decomposition), you read off dominant oscillation modes and decay constants. Schmid’s DMD with Control has been applied to PIV data of Kármán vortex streets.

When to use which:

• Clean data, interpretable variables, want an equation → SINDy.
• Noisy data, want forecasting/control, equation form irrelevant → Koopman/DMD.
• Both → SINDy on a Koopman-invariant subspace (Otto & Rowley, 2019).

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.