View source for Hamilton's principle/History/d'Alembert and virtual work

{{DISPLAYTITLE: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 ==

[[File:OPL-HP-F-18.png|thumb|right|400px|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:

:<math>\sum_i \mathbf{F}_i \cdot \delta \mathbf{r}_i = 0
\qquad\qquad\mathrm{(1)}</math>

The displacements <math>\delta\mathbf{r}_i</math> 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|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 ==

[[File:OPL-HP-F-33.png|thumb|right|400px|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
<math>-m_i\ddot{\mathbf{r}}_i</math> as a force. Then

:<math>\sum_i \bigl( \mathbf{F}_i - m_i\ddot{\mathbf{r}}_i \bigr)
\cdot \delta \mathbf{r}_i = 0
\qquad\qquad\mathrm{(2)}</math>

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

:<math>\Sigma\, . \, m(x''\,\delta x + y''\,\delta y + z''\,\delta z) = \delta U
\qquad\qquad\mathrm{(3)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (1.).
Verbatim. ESTABLISHED source. This is the volume's only verbatim text on this
page, and it is Hamilton's, not d'Alembert's.</ref>

Equation (3) '''is''' (2) with the applied forces derived from a force-function
<math>U</math>. 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|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:

:<math>-\int_{t_0}^{t_1} \sum_i m_i\ddot{\mathbf{r}}_i \cdot \delta\mathbf{r}_i\, dt
= \left[ -\sum_i m_i\dot{\mathbf{r}}_i \cdot \delta\mathbf{r}_i \right]_{t_0}^{t_1}
+ \int_{t_0}^{t_1} \sum_i m_i \dot{\mathbf{r}}_i \cdot \delta\dot{\mathbf{r}}_i\, dt
\qquad\qquad\mathrm{(4)}</math>

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 first variation]]. The remaining integrand
is <math>\delta T</math>. If the applied forces come from a potential, the first
term of (2) integrates to <math>-\delta V</math>, and the whole becomes

:<math>\int_{t_0}^{t_1} \delta\bigl( T - V \bigr)\, dt = 0
\qquad\qquad\mathrm{(5)}</math>

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|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|Lagrange and the Mecanique analytique]]. Hamilton
begins from the combined formula and never re-derives it.

<blockquote>
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|Sources]].
</blockquote>

== References ==

<references />