Hamilton's principle/Mathematics/Several dependent variables

Plain-language version: More than one variable

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

Setup

Varying one degree of freedom at a time.

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

$y_i(a)=y_{i,a},y_i(b)=y_{i,b},i=1,\dots ,n\mathrm{(}\mathrm{1}\mathrm{)}$

Let $L(x,y_1,\dots ,y_n,{y_1}^{\prime },\dots ,{y_n}^{\prime })$ have continuous second partials in all $2n+1$ slots, and define

$J[y]={\int }_a^{b}L(x,y_i,{y_i}^{\prime })dx\mathrm{(}\mathrm{2}\mathrm{)}$

The variation space is now $n$ copies of the earlier one: each ${\eta }_i\in C^2[a,b]$ with ${\eta }_i(a)={\eta }_i(b)=0$, and the ${\eta }_i$ are independent of one another. That independence is the whole content of what follows.

Derivation

Form the family $y_i+\epsilon {\eta }_i$, differentiate under the integral, and set $\epsilon =0$:

$\delta J={\int }_a^b\sum _{i=1}^n[\frac{\partial L}{\partial y_i}{\eta }_i+\frac{\partial L}{\partial {y_i}^{\prime }}{{\eta }_i}^{\prime }]dx\mathrm{(}\mathrm{3}\mathrm{)}$

Integrate each term of the second kind by parts. Every boundary term vanishes, since every ${\eta }_i$ vanishes at both ends:

$\delta J={\int }_a^b\sum _{i=1}^n[\frac{\partial L}{\partial y_i}-\frac{d}{dx}\frac{\partial L}{\partial {y_i}^{\prime }}]{\eta }_{i}dx\mathrm{(}\mathrm{4}\mathrm{)}$

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

Select one index at a time. Fix $k$ and choose ${\eta }_i=0$ for all $i\ne k$, with ${\eta }_k$ arbitrary. This is legitimate precisely because the ${\eta }_i$ are independent. The sum in (4) collapses to its $k$-th term, and the fundamental lemma of Hamilton's principle/Mathematics/The fundamental lemma gives

$\frac{\partial L}{\partial y_k}-\frac{d}{dx}\frac{\partial L}{\partial {y_k}^{\prime }}=0\mathrm{(}\mathrm{5}\mathrm{)}$

Repeating for each $k$ yields $n$ equations. $◼$

The single-index trick fails under constraints. If the $y_i$ are related — a bead confined to a wire, a rigid rod between two masses — then the ${\eta }_i$ are not independent, ${\eta }_k$ 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 $n$ coupled second-order equations, so its general solution carries $2n$ constants. The endpoint data (1) supplies exactly $2n$ conditions. The count matches again, as it did for one variable.

Coupling enters through the mixed partials ${\partial }^{2}L/\partial {y_i}^{\prime }\partial {y_k}^{\prime }$. Expanding the total derivative in (5),

$\sum _{k=1}^n\frac{{\partial }^{2}L}{\partial {y_i}^{\prime }\partial {y_k}^{\prime }}{y_k}^{″}+\sum _{k=1}^n\frac{{\partial }^{2}L}{\partial {y_i}^{\prime }\partial y_k}{y_k}^{\prime }+\frac{{\partial }^{2}L}{\partial {y_i}^{\prime }\partial x}-\frac{\partial L}{\partial y_i}=0\mathrm{(}\mathrm{6}\mathrm{)}$

The matrix $W_{ik}={\partial }^{2}L/\partial {y_i}^{\prime }\partial {y_k}^{\prime }$ is the Hessian with respect to the derivatives. The system can be solved for the ${y_k}^{″}$ if and only if $W$ is invertible. This is the $n$-variable non-degeneracy condition, and its definiteness version is the Legendre condition of Hamilton's principle/Mathematics/The Legendre condition. A singular $W$ 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 $L$ does not contain $y_k$, then

$p_k=\frac{\partial L}{\partial {y_k}^{\prime }}=\text{const}\mathrm{(}\mathrm{7}\mathrm{)}$

and $y_k$ is called a cyclic or ignorable coordinate.

If $L$ does not contain $x$, then

$\sum _{i=1}^n{y_i}^{\prime }\frac{\partial L}{\partial {y_i}^{\prime }}-L=\text{const}\mathrm{(}\mathrm{8}\mathrm{)}$

which is the $n$-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,

$\Sigma .m(x^{″}\delta x+y^{″}\delta y+z^{″}\delta z)=\delta U\mathrm{(}\mathrm{9}\mathrm{)}$

and after transforming to general marks of position ${\eta }_1,\dots ,{\eta }_{3n}$ he arrives at

$\frac{d}{dt}\frac{\delta T}{\delta {\eta }^{\prime }}-\frac{\delta T}{\delta \eta }=\frac{\delta U}{\delta \eta }\mathrm{(}\mathrm{10}\mathrm{)}$

Equation (10) is (5) with $L=T+U$, 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 $x$ as time $t$, coordinates $(y_1,y_2)$, and

$L=\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)\mathrm{(}\mathrm{11}\mathrm{)}$

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

$t$ does not appear either, so (8) applies as well and gives $\frac{1}{2}m({\dot{y}}_1^2+{\dot{y}}_2^2)=\text{const}$. 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.