Hamilton's principle/History/d'Alembert and virtual work

Between Euler's Methodus inveniendi of 1744 and Lagrange's Mécanique analytique of 1788 lies a development that is not a variational principle at all, and which the variational principles had to absorb before they could describe constrained systems. It is the principle of virtual work, and its extension to moving systems by d'Alembert.

This page carries no quotation. It is placed here because the material is necessary to the story and because the two principles it connects are treated elsewhere in the volume from their own sources.

Virtual work in statics

A constraint force normal to the permitted motion.

The principle of virtual work states that a mechanical system is in equilibrium if and only if the total work done by the applied forces vanishes for every displacement compatible with the constraints:

$\sum _i{𝐅}_i\cdot \delta {𝐫}_i=0\mathrm{(}\mathrm{1}\mathrm{)}$

The displacements $\delta {𝐫}_i$ are virtual: imagined, instantaneous, and consistent with the constraints. They are not motions the system undergoes.

The content of (1) is what it leaves out. Constraint forces do no virtual work, because they are normal to the constraint surface and the virtual displacements are tangent to it. So (1) contains only the applied forces, and the constraint forces — the unknown tensions and normal reactions that make a constrained problem hard — are eliminated before any calculation begins.

That elimination is the whole value of the principle, and it is the same observation that makes the multiplier method work on Hamilton's principle/Mathematics/Lagrange multipliers: because constraint forces do no work under admissible variations, the Euler-Lagrange expression can be nonzero and orthogonal to every admissible variation without contradiction.

D'Alembert's extension

D'Alembert's principle: dynamics rewritten as statics.

Statics is a special case of dynamics in which the accelerations vanish. D'Alembert's step, published in his Traité de dynamique of 1743, was to move the accelerations to the other side of the equation and treat $-m_i{\ddot{𝐫}}_i$ as a force. Then

$\sum _i({𝐅}_i-m_i{\ddot{𝐫}}_i)\cdot \delta {𝐫}_i=0\mathrm{(}\mathrm{2}\mathrm{)}$

for every virtual displacement compatible with the constraints.

(2) reduces every dynamical problem to a statical one. A moving system is in equilibrium under the applied forces together with the inertial forces, and the whole apparatus of virtual work applies unchanged.

The reformulation looks slight and is not. Newton's laws are statements about individual particles and require the constraint forces to be named. Equation (2) is a statement about the system as a whole and never names them.

Compare Hamilton's formula

Hamilton opens the 1834 essay by writing the equations of motion of a free system in the single formula

$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U\mathrm{(}\mathrm{3}\mathrm{)}$

Equation (3) is (2) with the applied forces derived from a force-function $U$. Hamilton takes the d'Alembert formula as his starting point and builds the characteristic function from it; the derivation on Hamilton's principle/Mathematics/Action as a function of its endpoints begins exactly there.

So the line from d'Alembert to Hamilton is direct, and it runs through a statement that is not itself variational.

From virtual work to a variational principle

Equation (2) is a condition at each instant. Hamilton's principle is a condition on a whole path. Getting from one to the other is an integration, and the step is worth seeing because it shows what the variational form adds.

Integrate (2) over time between fixed endpoints. The inertial term integrates by parts:

$-{\int }_{t_0}^{t_1}\sum _i{m}_i{\ddot{𝐫}}_i\cdot \delta {𝐫}_{i}dt={[-\sum _i{m}_i{\dot{𝐫}}_i\cdot \delta {𝐫}_i]}_{t_0}^{t_1}+{\int }_{t_0}^{t_1}\sum _i{m}_i{\dot{𝐫}}_i\cdot \delta {\dot{𝐫}}_{i}dt\mathrm{(}\mathrm{4}\mathrm{)}$

The boundary term vanishes because the variations vanish at the endpoints — the same step, and the same reason, as in Hamilton's principle/Mathematics/The first variation. The remaining integrand is $\delta T$. If the applied forces come from a potential, the first term of (2) integrates to $-\delta V$, and the whole becomes

${\int }_{t_0}^{t_1}\delta (T-V)dt=0\mathrm{(}\mathrm{5}\mathrm{)}$

which is Hamilton's principle.

What the integration buys. Equation (2) requires a vector statement at every instant. Equation (5) requires a single scalar functional to be stationary. The constraints, already absent from (2), stay absent, and the choice of coordinates becomes free — which is why the variational form generalises to field theory and the virtual-work form does not.

What it costs. The step from (2) to (5) assumed the forces derive from a potential. Virtual work needs no such assumption and applies to friction and other dissipative forces, where Hamilton's principle in its standard form does not. D'Alembert's principle is the more general statement and the less powerful tool.

Non-holonomic constraints

Virtual work makes contact with the warning issued on Hamilton's principle/Mathematics/Holonomic constraints.

For constraints on velocities that do not integrate to relations among coordinates, the correct dynamics comes from applying (2) to the velocity constraint. Adjoining the constraint to the action with a multiplier and varying gives a different and generally wrong answer.

The two methods agree for holonomic constraints and disagree for non-holonomic ones. That disagreement is the sharpest reason not to treat d'Alembert's principle as merely a step on the road to the variational one: for the rolling body it is the correct statement, and the variational statement is not.

Placement in the story

D'Alembert's contribution is not a variational principle and he did not intend one. His Traité predates the Methodus inveniendi by a year and was written in a different tradition.

What he supplied was the technique for handling constrained systems that the variational programme needed. Lagrange combined it with the calculus of variations, and the Mécanique analytique is the result; that is Hamilton's principle/History/Lagrange and the Mecanique analytique. Hamilton begins from the combined formula and never re-derives it.

Provenance: this page quotes no primary source except Hamilton's equation (1.), which is ESTABLISHED. D'Alembert's Traité de dynamique is not a source of this volume and no words or equations are attributed to him; equations (1), (2) and (4) are standard and are stated here. See Hamilton's principle/Sources.