BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:57:42Z

Pipe internal links so cross-references read as titles, not paths

← Older revision		Revision as of 04:57, 16 September 2026
Line 7:		Line 7:
	joined by an inextensible string — in each the coordinates are tied together, and		joined by an inextensible string — in each the coordinates are tied together, and
	the select-one-index argument of		the select-one-index argument of
	[[Hamilton's principle/Mathematics/Several dependent variables]] fails at its		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] fails at its
	first step. This page states the failure exactly. The repair is on		first step. This page states the failure exactly. The repair is on
	[[Hamilton's principle/Mathematics/Lagrange multipliers]].		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]].

	== Definition ==		== Definition ==
Line 41:		Line 41:
	'''generalised coordinates'''. In terms of them the problem is unconstrained, the		'''generalised coordinates'''. In terms of them the problem is unconstrained, the
	variations are again independent, and		variations are again independent, and
	[[Hamilton's principle/Mathematics/Several dependent variables]] applies		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] applies
	verbatim.		verbatim.

Line 75:		Line 75:
	So the step that produced <math>n</math> separate equations is unavailable. What		So the step that produced <math>n</math> separate equations is unavailable. What
	survives from		survives from
	[[Hamilton's principle/Mathematics/Several dependent variables]] is only the		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]] is only the
	integral statement		integral statement

Line 108:		Line 108:

	with the right side the constraint force. Equation (5) is what		with the right side the constraint force. Equation (5) is what
	[[Hamilton's principle/Mathematics/Lagrange multipliers]] derives; it is stated		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]] derives; it is stated
	here only to name the target.		here only to name the target.

Line 169:		Line 169:
	Provenance: the vakonomic-versus-dynamic distinction is editorial summary of the		Provenance: the vakonomic-versus-dynamic distinction is editorial summary of the
	modern literature. No primary source is cited for it and none is claimed. See		modern literature. No primary source is cited for it and none is claimed. See
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Lagrange multipliers]] converts the		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]] converts the
	restricted statement (4) with (3) into the unrestricted system (5).		restricted statement (4) with (3) into the unrestricted system (5).

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:46:34Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:46, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Holonomic constraints}}
	{{Plebeian\|Hamilton's principle/Explainers/Constraints}}		{{Plebeian\|Hamilton's principle/Explainers/Constraints}}

Line 179:		Line 180:

	<references />		<references />

BFP Rauch: 1 revision imported

2026-09-09T05:05:13Z

1 revision imported

← Older revision	Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of M2-01

2026-09-08T12:00:00Z

Initial import of M2-01

New page

{{Plebeian|Hamilton's principle/Explainers/Constraints}}

Every derivation in the machinery track rested on one assumption: that the
variations of the several unknowns could be chosen independently. Real systems
violate it. A bead threaded on a wire, a pendulum on a rigid rod, two masses
joined by an inextensible string — in each the coordinates are tied together, and
the select-one-index argument of
[[Hamilton's principle/Mathematics/Several dependent variables]] fails at its
first step. This page states the failure exactly. The repair is on
[[Hamilton's principle/Mathematics/Lagrange multipliers]].

== Definition ==

[[File:OPL-HP-F-18.png|thumb|right|400px|A constraint force normal to the permitted motion.]]

A '''holonomic constraint''' on <math>n</math> coordinates
<math>y_1, \ldots, y_n</math> is a relation

:<math>g\bigl(x,\, y_1, \ldots, y_n\bigr) = 0
\qquad\qquad\mathrm{(1)}</math>

involving the coordinates and possibly the independent variable, but '''not'''
their derivatives. A system of <math>r</math> such relations
<math>g_\beta = 0</math>, <math>\beta = 1, \ldots, r</math>, is holonomic if
each has this form and the <math>r \times n</math> matrix
<math>\partial g_\beta / \partial y_i</math> has rank <math>r</math> everywhere
on the constraint set. That rank condition says the constraints are independent
and non-degenerate; it is assumed throughout.

A constraint that cannot be written in the form (1) is '''non-holonomic'''. The
distinction is not cosmetic, and it is drawn at the end of this page.

== Two ways to proceed ==

[[File:OPL-HP-F-44.png|thumb|right|400px|Varying one degree of freedom at a time.]]

'''Eliminate.''' With <math>r</math> independent relations, <math>r</math> of the
coordinates can in principle be solved for and substituted out, leaving
<math>n - r</math> free coordinates <math>q_1, \ldots, q_{n-r}</math>. These are
'''generalised coordinates'''. In terms of them the problem is unconstrained, the
variations are again independent, and
[[Hamilton's principle/Mathematics/Several dependent variables]] applies
verbatim.

'''Adjoin.''' Keep all <math>n</math> coordinates and carry the constraints
alongside, paying for them with <math>r</math> new unknowns. This is the
multiplier method.

Elimination is cheaper when it is possible. It is often not possible in closed
form, it usually destroys symmetry, and it discards the constraint forces, which
are frequently the quantities of interest. The multiplier method keeps all three
and is the general tool.

== Why the independence argument fails ==

Let <math>y</math> be an admissible path satisfying (1), and let
<math>y + \varepsilon\eta</math> be a competitor. For the competitor to be
admissible it must satisfy the constraint too:

:<math>g\bigl(x,\, y + \varepsilon\eta\bigr) = 0
\qquad\qquad\mathrm{(2)}</math>

Expanding to first order in <math>\varepsilon</math> and using
<math>g(x,y) = 0</math>,

:<math>\sum_{i=1}^n \frac{\partial g}{\partial y_i}\, \eta_i = 0
\qquad\qquad\mathrm{(3)}</math>

Equation (3) is the '''admissibility condition on the variations'''. It is a
linear relation among the <math>\eta_i</math> at each point <math>x</math>.
Choosing <math>\eta_k</math> arbitrary and all others zero violates it, unless
<math>\partial g / \partial y_k</math> happens to vanish there.

So the step that produced <math>n</math> separate equations is unavailable. What
survives from
[[Hamilton's principle/Mathematics/Several dependent variables]] is only the
integral statement

:<math>\int_a^b \sum_{i=1}^n E_i\, \eta_i \, dx = 0,
\qquad
E_i = \frac{\partial L}{\partial y_i}
- \frac{d}{dx}\frac{\partial L}{\partial y_i'}
\qquad\qquad\mathrm{(4)}</math>

holding for those <math>\eta</math> obeying (3), and no longer for all
<math>\eta</math>. The fundamental lemma does not apply, because its hypothesis
is that the test class is unrestricted.

This is the whole difficulty, and it is a genuine one rather than a technicality.
Equation (4) with (3) says the vector <math>E</math> is orthogonal to every
admissible variation. It does not say <math>E</math> is zero. It says
<math>E</math> lies in the space spanned by the constraint gradients, and turning
that observation into a usable equation is what multipliers do.

== Constraint forces ==

Physically, <math>E_i \neq 0</math> is expected. The unconstrained
Euler-Lagrange expression measures the mismatch between the applied forces and
the motion, and for a constrained system that mismatch is exactly the force the
constraint exerts. The wire pushes on the bead. The rod pulls on the bob.

So the constrained problem should be expected to read

:<math>E_i = -\sum_{\beta=1}^r \lambda_\beta\,
\frac{\partial g_\beta}{\partial y_i}
\qquad\qquad\mathrm{(5)}</math>

with the right side the constraint force. Equation (5) is what
[[Hamilton's principle/Mathematics/Lagrange multipliers]] derives; it is stated
here only to name the target.

Note the structure: constraint forces are normal to the constraint surface, so
they do no work under any admissible variation. That is d'Alembert's principle,
and it is why (4) can hold with <math>E \neq 0</math> without contradiction.

== Worked example: the simple pendulum ==

A bob of mass <math>m</math> on a rigid rod of length <math>\ell</math>, in the
plane, under gravity <math>g_0</math>. Cartesian coordinates
<math>(y_1, y_2)</math> measured from the pivot, with <math>y_2</math> downward.

:<math>L = \tfrac{1}{2}m\bigl(\dot y_1^2 + \dot y_2^2\bigr) + m g_0 y_2,
\qquad
g = y_1^2 + y_2^2 - \ell^2 = 0
\qquad\qquad\mathrm{(6)}</math>

'''By elimination.''' Put <math>y_1 = \ell\sin\theta</math>,
<math>y_2 = \ell\cos\theta</math>. The constraint is satisfied identically, one
free coordinate remains, and

:<math>L = \tfrac{1}{2}m\ell^2\dot\theta^2 + m g_0 \ell\cos\theta
\qquad\qquad\mathrm{(7)}</math>

The single Euler-Lagrange equation is

:<math>m\ell^2\ddot\theta + m g_0 \ell \sin\theta = 0
\qquad\qquad\mathrm{(8)}</math>

which is the pendulum equation. Nothing about the rod tension appears, and it
cannot be recovered from (8) alone.

'''What was given up.''' The elimination worked because the constraint was
solvable in closed form and the parametrisation was obvious. Neither is generally
true. The multiplier treatment of the same system, on the next page, returns (8)
together with the tension.

== Non-holonomic constraints ==

A constraint involving derivatives,

:<math>\sum_{i=1}^n a_i(x,y)\, y_i' + a_0(x,y) = 0
\qquad\qquad\mathrm{(9)}</math>

is holonomic in disguise if the left side is an exact derivative or has an
integrating factor, in which case it integrates to the form (1). Otherwise it is
genuinely non-holonomic. The rolling coin on a plane is the standard example: it
can reach any position and orientation, so the constraint removes no
configurations, yet it restricts every motion.

'''Warning.''' Naively adjoining a non-holonomic constraint with a multiplier and
varying gives the '''vakonomic''' equations, which are ''not'' the equations of
the rolling body. The correct dynamics comes from d'Alembert's principle applied
to the velocity constraint, and the two disagree. This volume treats holonomic
constraints only, and records the distinction here so that the multiplier method
of the next page is not over-applied.

<blockquote>
Provenance: the vakonomic-versus-dynamic distinction is editorial summary of the
modern literature. No primary source is cited for it and none is claimed. See
[[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Mathematics/Lagrange multipliers]] converts the
restricted statement (4) with (3) into the unrestricted system (5).

== References ==

<references />

Hamilton's principle/Mathematics/Holonomic constraints - Revision history

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported

hamiltons-principle>HPVBuild: Initial import of M2-01