import numpy as np
import sympy as sym
import matplotlib as mpl
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
from IPython.display import Math, display
mpl.rcParams['figure.dpi'] = 150
mpl.rcParams['axes.spines.top'] = False
mpl.rcParams['axes.spines.right'] = FalseThis section covers the basics of dimensional analysis and the process of non-dimensionalizing a differential equation. These are powerful and broadly applicable techniques that:
These ideas are standard tools in applied mathematics, physics, and engineering, and they appear throughout the analysis of ODEs. Good introductory references are Logan (2015) (Section 1.2), Strogatz (2015) (Section 3.5), and the Wikipedia article on nondimensionalization.
Every measurable physical quantity has a dimension (or unit dimension), that is, a qualitative description of what kind of thing is being measured. This is typically known as a unit which is a conventional standard for expressing that dimension numerically.
In mechanics the three fundamental dimensions are:
| Symbol | Dimension |
|---|---|
| \(\mathsf{M}\) | Mass |
| \(\mathsf{L}\) | Length |
| \(\mathsf{T}\) | Time |
Every other mechanical quantity is expressed as a product of powers of these. We write \([Q]\) for the dimension of quantity \(Q\). For example:
| Quantity | Dimension | SI Unit |
|---|---|---|
| Time \(t\) | \(\mathsf{T}\) | second (s) |
| Length/position \(x\) | \(\mathsf{L}\) | metre (m) |
| Velocity \(v = dx/dt\) | \(\mathsf{L}\,\mathsf{T}^{-1}\) | m s\(^{-1}\) |
| Acceleration \(a = d^2x/dt^2\) | \(\mathsf{L}\,\mathsf{T}^{-2}\) | m s\(^{-2}\) |
| Mass \(m\) | \(\mathsf{M}\) | kilogram (kg) |
| Force \(F = ma\) | \(\mathsf{M}\,\mathsf{L}\,\mathsf{T}^{-2}\) | newton (N) |
| Energy \(E\) | \(\mathsf{M}\,\mathsf{L}^{2}\,\mathsf{T}^{-2}\) | joule (J) |
| Population \(P\) | dimensionless | individuals |
| Rate constant \(r\) | \(\mathsf{T}^{-1}\) | per second |
Every term in a physically meaningful equation must have identical dimensions. An equation that adds a length to a time, for instance, is dimensionally inconsistent and cannot be correct.
This principle is both a consistency check and a discovery tool: if a proposed formula is dimensionally inconsistent, it is wrong; if you know the dimensions of the answer, you can often guess the form of the formula.
Suppose a pendulum of length \(\ell\) swings under gravity \(g\) (with \([g] = \mathsf{L}\,\mathsf{T}^{-2}\)). What is the period \(\tau\)?
We expect \(\tau = C\,\ell^a g^b\) for some pure number \(C\) and unknown exponents \(a, b\). Matching dimensions: \[ [\tau] = \mathsf{T} = \mathsf{L}^a \cdot (\mathsf{L}\,\mathsf{T}^{-2})^b = \mathsf{L}^{a+b}\,\mathsf{T}^{-2b}. \] Equating exponents: \(-2b = 1 \Rightarrow b = -\tfrac{1}{2}\); \(a + b = 0 \Rightarrow a = \tfrac{1}{2}\). Therefore \[ \tau = C\sqrt{\frac{\ell}{g}}. \] Dimensional analysis alone tells us the \(\sqrt{\ell/g}\) scaling; a full solution of the pendulum ODE gives \(C = 2\pi\).
The systematic foundation for dimensional analysis is the Buckingham \(\Pi\) theorem:
If a physical relationship involves \(n\) dimensional quantities built from \(k\) independent fundamental dimensions, then it can be expressed as a relationship among \(n - k\) dimensionless groups \(\Pi_1, \Pi_2, \ldots, \Pi_{n-k}\).
Each \(\Pi_i\) is a product of powers of the original quantities chosen so that all dimensions cancel. The theorem guarantees that reducing \(n\) parameters to \(n-k\) dimensionless groups is always possible and that nothing is lost in doing so.
Non-dimensionalization is the process of rescaling the dependent and independent variables of a differential equation so that all variables and parameters become dimensionless. The procedure is:
We now apply this procedure to two important examples.
The logistic equation for a population \(P(t)\) is \[ \frac{dP}{dt} = rP\!\left(1 - \frac{P}{K}\right), \qquad P(0) = P_0, \] where:
| Parameter | Meaning | Dimension |
|---|---|---|
| \(P\) | population size | \(\mathsf{N}\) (number of individuals) |
| \(t\) | time | \(\mathsf{T}\) |
| \(r > 0\) | intrinsic growth rate | \(\mathsf{T}^{-1}\) |
| \(K > 0\) | carrying capacity | \(\mathsf{N}\) |
| \(P_0\) | initial population | \(\mathsf{N}\) |
Here we use \(\mathsf{N}\) (number) as a fourth dimension for population counts, though it is often treated as dimensionless in practice.
The equation has three dimensional parameters: \(r\), \(K\), \(P_0\).
Define \[ u = \frac{P}{K}, \qquad \tau = r\,t. \] Both \(u\) and \(\tau\) are dimensionless by construction. Note that \(P = Ku\) and \(t = \tau/r\), so \[ \frac{dP}{dt} = \frac{d(Ku)}{d(\tau/r)} = Kr\,\frac{du}{d\tau}. \]
\[ Kr\,\frac{du}{d\tau} = r\,(Ku)\!\left(1 - \frac{Ku}{K}\right) = rKu(1 - u). \] Dividing both sides by \(Kr\): \[ \boxed{\frac{du}{d\tau} = u(1-u), \qquad u(0) = u_0 \equiv \frac{P_0}{K}.} \]
The original ODE had three parameters \((r, K, P_0)\). The non-dimensionalized ODE has one parameter \(u_0 = P_0/K\) — the initial population expressed as a fraction of the carrying capacity. The parameters \(r\) and \(K\) have been entirely absorbed into the rescaled variables; they set the scales but do not alter the shape of the solution.
This means that two logistic populations with different values of \(r\) and \(K\) but the same ratio \(P_0/K\) follow exactly the same dimensionless trajectory. Non-dimensionalization makes this universality explicit.
The solution in dimensionless variables (from our earlier work) is \[ u(\tau) = \frac{u_0}{u_0 + (1 - u_0)e^{-\tau}}, \] and converting back: \(P(t) = K\,u(rt)\).
# Three different (r, K, P0) sets that share the same u0 = P0/K = 0.1
configs = [
dict(r=1.0, K=100, P0=10, label=r'$r=1,\;K=100,\;P_0=10$', color='steelblue'),
dict(r=0.5, K=200, P0=20, label=r'$r=0.5,\;K=200,\;P_0=20$', color='crimson'),
dict(r=2.0, K=50, P0=5, label=r'$r=2,\;K=50,\;P_0=5$', color='seagreen'),
]
fig, axes = plt.subplots(1, 2, figsize=(11, 4))
for cfg in configs:
r, K, P0 = cfg['r'], cfg['K'], cfg['P0']
u0 = P0 / K
# Dimensional time axis: show ~5 intrinsic time scales
t_dim = np.linspace(0, 5 / r, 400)
P_sol = K * u0 / (u0 + (1 - u0) * np.exp(-r * t_dim))
axes[0].plot(t_dim, P_sol, lw=2, color=cfg['color'], label=cfg['label'])
# Dimensionless axis tau = r*t, same u0
tau = np.linspace(0, 10, 400)
u_sol = u0 / (u0 + (1 - u0) * np.exp(-tau))
axes[1].plot(tau, u_sol, lw=2, color=cfg['color'], label=cfg['label'])
axes[0].set_xlabel('$t$', fontsize=12)
axes[0].set_ylabel('$P(t)$', fontsize=12)
axes[0].set_title('Dimensional form', fontsize=12)
axes[0].legend(fontsize=8)
axes[1].set_xlabel(r'$\tau = rt$', fontsize=12)
axes[1].set_ylabel(r'$u(\tau) = P/K$', fontsize=12)
axes[1].set_title('Dimensionless form', fontsize=12)
axes[1].axhline(1, color='black', linestyle='--', lw=1, label='$u=1$ (carrying capacity)')
axes[1].legend(fontsize=8)
plt.suptitle('Logistic Growth: Dimensional vs. Dimensionless', fontsize=13)
plt.tight_layout()
plt.show()
The equation of motion for a damped harmonic oscillator driven by a periodic force is \[ \frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2\,x = \frac{F_0}{m}\cos(\Omega t), \] where (using SI units for concreteness):
| Parameter | Meaning | Dimension |
|---|---|---|
| \(x\) | displacement | \(\mathsf{L}\) |
| \(t\) | time | \(\mathsf{T}\) |
| \(\gamma > 0\) | damping coefficient (per unit mass) | \(\mathsf{T}^{-1}\) |
| \(\omega_0 > 0\) | natural angular frequency | \(\mathsf{T}^{-1}\) |
| \(F_0/m\) | forcing amplitude (per unit mass) | \(\mathsf{L}\,\mathsf{T}^{-2}\) |
| \(\Omega\) | driving frequency | \(\mathsf{T}^{-1}\) |
We write the damping term as \(2\gamma\) (rather than \(\gamma\)) purely as a notational convenience that simplifies the algebra later; \(\gamma\) is sometimes called the damping rate or decay rate.
The equation has four dimensional parameters: \(\gamma\), \(\omega_0\), \(F_0/m\), \(\Omega\).
Define \[ u = \frac{x}{\mathcal{X}} = \frac{x\,\omega_0^2}{F_0/m}, \qquad \tau = \omega_0\,t. \] Then \(x = \mathcal{X}\,u\) and \(t = \tau/\omega_0\). Computing derivatives via the chain rule: \[ \frac{dx}{dt} = \mathcal{X}\,\omega_0\,\frac{du}{d\tau}, \qquad \frac{d^2x}{dt^2} = \mathcal{X}\,\omega_0^2\,\frac{d^2u}{d\tau^2}. \]
Replace each term: \[ \mathcal{X}\omega_0^2\,\frac{d^2u}{d\tau^2} + 2\gamma\,\mathcal{X}\omega_0\,\frac{du}{d\tau} + \omega_0^2\,\mathcal{X}\,u = \frac{F_0}{m}\cos\!\left(\frac{\Omega}{\omega_0}\tau\right). \] Divide every term by \(\mathcal{X}\omega_0^2 = F_0/m\): \[ \boxed{\frac{d^2u}{d\tau^2} + 2\zeta\,\frac{du}{d\tau} + u = \cos(\nu\tau),} \] where we have introduced two key dimensionless groups:
\[ \zeta = \frac{\gamma}{\omega_0} \quad \text{(damping ratio)}, \qquad \nu = \frac{\Omega}{\omega_0} \quad \text{(frequency ratio)}. \]
The original ODE had four parameters \((\gamma, \omega_0, F_0/m, \Omega)\). After non-dimensionalization only two remain:
| Group | Formula | Physical meaning |
|---|---|---|
| Damping ratio \(\zeta\) | \(\gamma/\omega_0\) | Ratio of damping rate to natural frequency |
| Frequency ratio \(\nu\) | \(\Omega/\omega_0\) | Ratio of driving frequency to natural frequency |
The forcing amplitude \(F_0/m\) and natural frequency \(\omega_0\) have been absorbed entirely into the scales \(\mathcal{X}\) and \(\mathcal{T}\).
The dimensionless equation makes resonance transparent: the system is driven at its natural frequency when \(\nu = \Omega/\omega_0 = 1\). The concept of resonance is a statement about the ratio of frequencies, not their individual values — a fact that non-dimensionalization makes explicit.
The three classical damping regimes are determined entirely by \(\zeta\):
| \(\zeta\) | Regime | Behavior |
|---|---|---|
| \(0 < \zeta < 1\) | Underdamped | Oscillatory decay toward steady state |
| \(\zeta = 1\) | Critically damped | Fastest non-oscillatory return to equilibrium |
| \(\zeta > 1\) | Overdamped | Slow exponential return, no oscillation |
For the non-dimensionalized driven oscillator, the steady-state (particular) solution has the form \[ u_p(\tau) = R(\nu,\zeta)\cos(\nu\tau - \phi), \] where the amplitude response function and phase are \[ R(\nu,\zeta) = \frac{1}{\sqrt{(1-\nu^2)^2 + 4\zeta^2\nu^2}}, \qquad \phi = \arctan\!\left(\frac{2\zeta\nu}{1-\nu^2}\right). \]
nu_vals = np.linspace(0, 2.5, 800)
zeta_list = [0.05, 0.1, 0.2, 0.5, 1.0]
colors = plt.cm.plasma(np.linspace(0.1, 0.85, len(zeta_list)))
fig, ax = plt.subplots(figsize=(8, 5))
for zeta, color in zip(zeta_list, colors):
R = 1.0 / np.sqrt((1 - nu_vals**2)**2 + 4 * zeta**2 * nu_vals**2)
ax.plot(nu_vals, R, lw=2, color=color, label=fr'$\zeta = {zeta}$')
ax.axvline(1, color='black', linestyle=':', lw=1.2, label=r'$\nu = 1$ (resonance)')
ax.set_xlabel(r'Frequency ratio $\nu = \Omega/\omega_0$', fontsize=12)
ax.set_ylabel(r'Amplitude $R(\nu,\zeta)$', fontsize=12)
ax.set_title('Amplitude Response of the Dimensionless Forced Oscillator', fontsize=12)
ax.set_ylim(0, 7)
ax.legend(fontsize=9)
plt.tight_layout()
plt.show()
The following plot shows full numerical solutions (transient + steady state) of the dimensionless ODE for several damping ratios at the resonant driving frequency \(\nu = 1\).
def forced_osc(tau, y, zeta, nu):
u, v = y
return [v, np.cos(nu * tau) - 2 * zeta * v - u]
tau_span = (0, 60)
tau_eval = np.linspace(*tau_span, 3000)
zeta_list2 = [0.05, 0.1, 0.2, 0.5]
colors2 = plt.cm.viridis(np.linspace(0.1, 0.85, len(zeta_list2)))
fig, ax = plt.subplots(figsize=(10, 4))
for zeta, color in zip(zeta_list2, colors2):
sol = solve_ivp(forced_osc, tau_span, [0, 0],
args=(zeta, 1.0), t_eval=tau_eval, max_step=0.05)
ax.plot(sol.t, sol.y[0], lw=1.5, color=color, label=fr'$\zeta={zeta}$')
ax.axhline(0, color='black', lw=0.5)
ax.set_xlabel(r'$\tau = \omega_0 t$', fontsize=12)
ax.set_ylabel(r'$u(\tau) = x / \mathcal{X}$', fontsize=12)
ax.set_title(r'Dimensionless Forced Oscillator at Resonance ($\nu=1$)', fontsize=12)
ax.legend(fontsize=9)
plt.tight_layout()
plt.show()
The table below collects the key results from both examples.
| ODE | Dimensional parameters | Dimensionless variables | Dimensionless groups |
|---|---|---|---|
| Logistic growth | \(r,\; K,\; P_0\) | \(u = P/K\), \(\;\tau = rt\) | \(u_0 = P_0/K\) |
| Forced oscillator | \(\gamma,\;\omega_0,\;F_0/m,\;\Omega\) | \(u = x\omega_0^2/(F_0/m)\), \(\;\tau=\omega_0 t\) | \(\zeta = \gamma/\omega_0\), \(\;\nu = \Omega/\omega_0\) |
import sys
print("Python version:", sys.version)
print('\n'.join(f'{m.__name__}=={m.__version__}' for m in globals().values() if getattr(m, '__version__', None)))Python version: 3.14.4 | packaged by conda-forge | (main, Apr 8 2026, 02:33:53) [Clang 20.1.8 ]
numpy==2.4.3
sympy==1.14.0
matplotlib==3.10.8