Hamilton's principle/Mathematics/Several dependent variables - Revision history

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	<math>\eta_k</math> arbitrary. This is legitimate precisely because the		<math>\eta_k</math> arbitrary. This is legitimate precisely because the
	<math>\eta_i</math> are independent. The sum in (4) collapses to its		<math>\eta_i</math> are independent. The sum in (4) collapses to its
	<math>k</math>-th term, and ~~the fundamental lemma of~~		<math>k</math>-th term, and [[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] gives
	[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] gives

	:<math>\frac{\partial L}{\partial y_k}		:<math>\frac{\partial L}{\partial y_k}

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			{{DISPLAYTITLE:Several dependent variables}}
	{{Plebeian\|Hamilton's principle/Explainers/More than one variable}}		{{Plebeian\|Hamilton's principle/Explainers/More than one variable}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/More than one variable}}

A mechanical system rarely has one degree of freedom. This page extends the
machinery to an integrand depending on <math>n</math> unknown functions of a
single independent variable. The result is a ''system'' of <math>n</math>
Euler-Lagrange equations, one per unknown, and this is the form in which
Hamilton's principle is actually stated for mechanics.

== Setup ==

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

Let <math>y = (y_1, \ldots, y_n)</math> be <math>n</math> functions of
<math>x</math> on <math>[a,b]</math>, each in <math>C^2</math>, with all
<math>2n</math> endpoint values prescribed:

:<math>y_i(a) = y_{i,a}, \qquad y_i(b) = y_{i,b},
\qquad i = 1, \ldots, n \qquad\qquad\mathrm{(1)}</math>

Let <math>L(x, y_1, \ldots, y_n, y_1', \ldots, y_n')</math> have continuous
second partials in all <math>2n+1</math> slots, and define

:<math>J[y] = \int_a^b L\bigl(x, y_i, y_i'\bigr)\, dx
\qquad\qquad\mathrm{(2)}</math>

The variation space is now <math>n</math> copies of the earlier one: each
<math>\eta_i \in C^2[a,b]</math> with
<math>\eta_i(a) = \eta_i(b) = 0</math>, and the <math>\eta_i</math> are
'''independent of one another'''. That independence is the whole content of what
follows.

== Derivation ==

Form the family <math>y_i + \varepsilon\eta_i</math>, differentiate under the
integral, and set <math>\varepsilon = 0</math>:

:<math>\delta J = \int_a^b \sum_{i=1}^n
\left[ \frac{\partial L}{\partial y_i}\,\eta_i
+ \frac{\partial L}{\partial y_i'}\,\eta_i' \right] dx
\qquad\qquad\mathrm{(3)}</math>

Integrate each term of the second kind by parts. Every boundary term vanishes,
since every <math>\eta_i</math> vanishes at both ends:

:<math>\delta J = \int_a^b \sum_{i=1}^n
\left[ \frac{\partial L}{\partial y_i}
- \frac{d}{dx}\frac{\partial L}{\partial y_i'} \right] \eta_i \, dx
\qquad\qquad\mathrm{(4)}</math>

Stationarity requires (4) to vanish for every choice of the <math>n</math>
variations. Here is the step that needs care.

'''Select one index at a time.''' Fix <math>k</math> and choose
<math>\eta_i = 0</math> for all <math>i \neq k</math>, with
<math>\eta_k</math> arbitrary. This is legitimate precisely because the
<math>\eta_i</math> are independent. The sum in (4) collapses to its
<math>k</math>-th term, and the fundamental lemma of
[[Hamilton's principle/Mathematics/The fundamental lemma]] gives

:<math>\frac{\partial L}{\partial y_k}
- \frac{d}{dx}\frac{\partial L}{\partial y_k'} = 0
\qquad\qquad\mathrm{(5)}</math>

Repeating for each <math>k</math> yields <math>n</math> equations.
<math>\blacksquare</math>

'''The single-index trick fails under constraints.''' If the <math>y_i</math> are
related — a bead confined to a wire, a rigid rod between two masses — then the
<math>\eta_i</math> are not independent, <math>\eta_k</math> cannot be varied
alone, and (5) does not follow. That case is
[[Hamilton's principle/Mathematics/Holonomic constraints]], and the repair is
[[Hamilton's principle/Mathematics/Lagrange multipliers]]. Everything in this
page rests on independence, and it is worth knowing exactly where.

== Counting ==

(5) is a system of <math>n</math> coupled second-order equations, so its general
solution carries <math>2n</math> constants. The endpoint data (1) supplies
exactly <math>2n</math> conditions. The count matches again, as it did for one
variable.

Coupling enters through the mixed partials
<math>\partial^2 L / \partial y_i' \partial y_k'</math>. Expanding the total
derivative in (5),

:<math>\sum_{k=1}^n \frac{\partial^2 L}{\partial y_i' \partial y_k'}\, y_k''
+ \sum_{k=1}^n \frac{\partial^2 L}{\partial y_i' \partial y_k}\, y_k'
+ \frac{\partial^2 L}{\partial y_i' \partial x}
- \frac{\partial L}{\partial y_i} = 0
\qquad\qquad\mathrm{(6)}</math>

The matrix <math>W_{ik} = \partial^2 L / \partial y_i' \partial y_k'</math> is
the '''Hessian''' with respect to the derivatives. The system can be solved for
the <math>y_k''</math> if and only if <math>W</math> is invertible. This is the
<math>n</math>-variable non-degeneracy condition, and its definiteness version is
the Legendre condition of
[[Hamilton's principle/Mathematics/The Legendre condition]]. A singular
<math>W</math> signals a constrained or gauge system, which is where
[[Hamilton's principle/Mathematics/Noether's second theorem]] applies.

== First integrals ==

Both results from the one-variable case generalise, with the same proofs.

If <math>L</math> does not contain <math>y_k</math>, then

:<math>p_k = \frac{\partial L}{\partial y_k'} = \text{const}
\qquad\qquad\mathrm{(7)}</math>

and <math>y_k</math> is called a '''cyclic''' or ignorable coordinate.

If <math>L</math> does not contain <math>x</math>, then

:<math>\sum_{i=1}^n y_i'\,\frac{\partial L}{\partial y_i'} - L = \text{const}
\qquad\qquad\mathrm{(8)}</math>

which is the <math>n</math>-variable Beltrami identity of
[[Hamilton's principle/Mathematics/The Beltrami identity]], and is the
Hamiltonian.

== Hamilton's own form ==

This is the form in which the historical statement is made. Hamilton writes the
equations of motion of a system of free points as a single formula,

:<math>\Sigma\, . \, m(x''\,\delta x + y''\,\delta y + z''\,\delta z) = \delta U
\qquad\qquad\mathrm{(9)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (1.).
Verbatim. The summation runs over the points of the system; <math>U</math> is
the force-function.</ref>

and after transforming to general marks of position <math>\eta_1, \ldots,
\eta_{3n}</math> he arrives at

:<math>\frac{d}{dt}\frac{\delta T}{\delta \eta'}
- \frac{\delta T}{\delta \eta} = \frac{\delta U}{\delta \eta}
\qquad\qquad\mathrm{(10)}</math>

<ref>Hamilton, ''On a General Method in Dynamics'' (1834), equation (Y.).
Verbatim in structure, with Hamilton's <math>\delta</math> used for partial
differentiation as he writes it. Hamilton remarks that these coincide, apart
from a slight difference of notation, with the canonical forms of Lagrange.</ref>

Equation (10) is (5) with <math>L = T + U</math>, allowing for Hamilton's sign
convention on the force-function. The correspondence between his notation and
this volume's is tabulated on [[Hamilton's principle/Notation]].

== Worked example: the free particle in the plane ==

Take <math>x</math> as time <math>t</math>, coordinates
<math>(y_1, y_2)</math>, and

:<math>L = \tfrac{1}{2}m\bigl( \dot{y}_1^2 + \dot{y}_2^2 \bigr)
\qquad\qquad\mathrm{(11)}</math>

Neither coordinate appears in <math>L</math>, so both are cyclic and (7) gives
<math>m\dot{y}_1 = c_1</math>, <math>m\dot{y}_2 = c_2</math>. Both momentum
components are conserved and the path is a straight line traversed at constant
speed.

<math>t</math> does not appear either, so (8) applies as well and gives
<math>\tfrac{1}{2}m(\dot{y}_1^2 + \dot{y}_2^2) = \text{const}</math>. That is
energy conservation, and it is not independent of the momentum results here —
it follows from them. Independent symmetries need not give independent first
integrals, a point Noether makes explicitly and which is taken up on
[[Hamilton's principle/Mathematics/Noether's first theorem]].

== Next ==

[[Hamilton's principle/Mathematics/Several independent variables]] takes the
remaining step, from ordinary to partial differential equations.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	<math>\eta_i</math> are independent. The sum in (4) collapses to its		<math>\eta_i</math> are independent. The sum in (4) collapses to its
	<math>k</math>-th term, and the fundamental lemma of		<math>k</math>-th term, and the fundamental lemma of
	[[Hamilton's principle/Mathematics/The fundamental lemma]] gives		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] gives

	:<math>\frac{\partial L}{\partial y_k}		:<math>\frac{\partial L}{\partial y_k}
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	<math>\eta_i</math> are not independent, <math>\eta_k</math> cannot be varied		<math>\eta_i</math> are not independent, <math>\eta_k</math> cannot be varied
	alone, and (5) does not follow. That case is		alone, and (5) does not follow. That case is
	[[Hamilton's principle/Mathematics/Holonomic constraints]], and the repair is		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]], and the repair is
	[[Hamilton's principle/Mathematics/Lagrange multipliers]]. Everything in this		[[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]]. Everything in this
	page rests on independence, and it is worth knowing exactly where.		page rests on independence, and it is worth knowing exactly where.

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	<math>n</math>-variable non-degeneracy condition, and its definiteness version is		<math>n</math>-variable non-degeneracy condition, and its definiteness version is
	the Legendre condition of		the Legendre condition of
	[[Hamilton's principle/Mathematics/The Legendre condition]]. A singular		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]]. A singular
	<math>W</math> signals a constrained or gauge system, which is where		<math>W</math> signals a constrained or gauge system, which is where
	[[Hamilton's principle/Mathematics/Noether's second theorem]] applies.		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] applies.

	== First integrals ==		== First integrals ==
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	which is the <math>n</math>-variable Beltrami identity of		which is the <math>n</math>-variable Beltrami identity of
	[[Hamilton's principle/Mathematics/The Beltrami identity]], and is the		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]], and is the
	Hamiltonian.		Hamiltonian.

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	Equation (10) is (5) with <math>L = T + U</math>, allowing for Hamilton's sign		Equation (10) is (5) with <math>L = T + U</math>, allowing for Hamilton's sign
	convention on the force-function. The correspondence between his notation and		convention on the force-function. The correspondence between his notation and
	this volume's is tabulated on [[Hamilton's principle/Notation]].		this volume's is tabulated on [[Hamilton's principle/Notation\|Notation]].

	== Worked example: the free particle in the plane ==		== Worked example: the free particle in the plane ==
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	it follows from them. Independent symmetries need not give independent first		it follows from them. Independent symmetries need not give independent first
	integrals, a point Noether makes explicitly and which is taken up on		integrals, a point Noether makes explicitly and which is taken up on
	[[Hamilton's principle/Mathematics/Noether's first theorem]].		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]].

	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Several independent variables]] takes the		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]] takes the
	remaining step, from ordinary to partial differential equations.		remaining step, from ordinary to partial differential equations.