✓ Sources checked · 10 September 2026 · View verification
D’Alembert’s principle writes dynamics as a virtual-work balance of applied forces and the inertial terms −miai. Those inertial terms are bookkeeping, not additional physical interactions.
By Matan · Updated
Dynamic virtual-work balance
For fixed particle masses and ideal constraints, Σ(Fi − miai) · δri = 0 for compatible virtual displacements. Fi denotes applied forces after constraint reactions that do no virtual work are eliminated. The acceleration is still the actual acceleration relative to the chosen inertial frame. [1]
Worked crate on a frictionless incline
A 2.0 kg crate slides on a fixed 30° slope. Choose positive displacement downhill. Gravity’s tangential component is mg sin 30° = 9.8 N for g = 9.8 m/s². The normal force is perpendicular to the allowed displacement, so it contributes nothing. Thus (9.8 − 2a)δs = 0 for arbitrary δs, giving a = 4.9 m/s² downhill.
Why this is not static equilibrium
The real tangential force is 9.8 N, not zero. Subtracting ma makes a mathematical balance while the crate accelerates. Treating −ma as a second contact force on a force diagram would double-count inertia.
Choose the model before using it
This approach leads naturally to Lagrange’s equations and helps eliminate ideal constraint forces. Dissipative forces must still be included when present. Rockets exchange mass and momentum with exhaust and need an appropriate system balance; the fixed-mass expression above is not a complete rocket equation.
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How this article was checked
Sources checked; calculations checked where shown. Main claims and model assumptions were checked against the references below. Example arithmetic and units were checked separately. Numerical examples are illustrative calculations, not experimental measurements.
Source checking and editing assisted by AI. No independent human scientific review is recorded for this revision. This record applies to the version dated 10 September 2026.
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Cite this article
Matan. “D’Alembert’s principle.” Your Physicist. Originally published 2023-03-17; updated 2026-09-10. https://your-physicist.com/dalemberts-principle/
Revision history
10 September 2026: Corrected the fictitious-force interpretation and supplied a constrained acceleration example. Added references and a transparent verification record.
