24 Jul 2025
Variational Principles, Constraints, and Least Action
Euler-Lagrange calculus, D'Alembert and modified Hamilton principles, multipliers, and constrained examples.
Calculus of variations and the Euler-Lagrange equation
Consider the functional
\[J[y]=\int_{x_1}^{x_2}F(x,y,y')\,dx.\]Vary the curve as $y(x,\epsilon)=y(x)+\epsilon\eta(x)$ while holding its endpoints fixed:
\[\eta(x_1)=\eta(x_2)=0.\]The first variation is
\[\delta J =\int_{x_1}^{x_2}\left( \frac{\partial F}{\partial y}\eta +\frac{\partial F}{\partial y'}\eta' \right)dx.\]After integration by parts,
\[\delta J =\left[\frac{\partial F}{\partial y'}\eta\right]_{x_1}^{x_2} +\int_{x_1}^{x_2}\left[ \frac{\partial F}{\partial y} -\frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) \right]\eta\,dx.\]The boundary term vanishes because the endpoints are fixed. Since $\eta$ is otherwise arbitrary, stationarity $\delta J=0$ requires
\[\boxed{ \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) -\frac{\partial F}{\partial y}=0}.\]For several dependent variables, the same equation holds for each one.
Hamilton’s principle from D’Alembert’s principle
D’Alembert’s equation in generalized coordinates is
\[\sum_j\left[ Q_j-\frac{d}{dt}\frac{\partial T}{\partial\dot q_j} +\frac{\partial T}{\partial q_j} \right]\delta q_j=0.\]For conservative forces $Q_j=-\partial V/\partial q_j$ and $L=T-V$, this becomes
\[\sum_j\left[ \frac{\partial L}{\partial q_j} -\frac{d}{dt}\frac{\partial L}{\partial\dot q_j} \right]\delta q_j=0.\]Integrate from $t_1$ to $t_2$. If the comparison paths have the same configurations at both endpoints,
\[\delta q_j(t_1)=\delta q_j(t_2)=0,\]then integration by parts gives
\[\boxed{ \delta S=0, \qquad S=\int_{t_1}^{t_2}L\,dt}.\]Thus Hamilton’s principle follows from D’Alembert’s principle for an ideal constrained conservative system.
Modified Hamilton principle
In phase space, regard $q_i(t)$ and $p_i(t)$ as independent and vary
\[S_H=\int_{t_1}^{t_2}\left(\sum_i p_i\dot q_i-H(q,p,t)\right)dt.\]Only the coordinate variations are fixed at the endpoints:
\[\delta q_i(t_1)=\delta q_i(t_2)=0;\]$\delta p_i$ need not vanish there. Direct variation and integration of $p_i\delta\dot q_i$ by parts give
\[\delta S_H =\left[\sum_i p_i\delta q_i\right]_{t_1}^{t_2} +\int_{t_1}^{t_2}\sum_i\left[ \left(\dot q_i-\frac{\partial H}{\partial p_i}\right)\delta p_i -\left(\dot p_i+\frac{\partial H}{\partial q_i}\right)\delta q_i \right]dt.\]The endpoint term is zero. Independent interior variations give both Hamilton equations.
Lagrange’s method of undetermined multipliers
For holonomic constraints
\[f_\alpha(q,t)=0, \qquad \alpha=1,\ldots,k,\]introduce multipliers $\lambda_\alpha(t)$ and the augmented Lagrangian
\[L_{\mathrm a}=L+\sum_\alpha\lambda_\alpha f_\alpha.\]Variation with respect to $q_i$ and $\lambda_\alpha$ yields
\[\boxed{ \frac{d}{dt}\frac{\partial L}{\partial\dot q_i} -\frac{\partial L}{\partial q_i} =\sum_\alpha\lambda_\alpha \frac{\partial f_\alpha}{\partial q_i}}, \qquad \boxed{f_\alpha=0}.\]The multiplier terms are the generalized constraint forces. The sign of a multiplier depends on whether $+\lambda_\alpha f_\alpha$ or $-\lambda_\alpha f_\alpha$ is chosen; physical forces do not depend on that convention.
Simple pendulum by a multiplier
Use Cartesian coordinates with $y$ positive upward:
\[L=\frac m2(\dot x^2+\dot y^2)-mgy, \qquad f=x^2+y^2-l^2=0.\]The multiplier equations are
\[m\ddot x=2\lambda x, \qquad m\ddot y=-mg+2\lambda y.\]Write $x=l\sin\theta$ and $y=-l\cos\theta$, with $\theta=0$ at the lowest point. Eliminating $\lambda$ gives
\[\boxed{\ddot\theta+\frac gl\sin\theta=0}.\]Because the constraint force is $2\lambda(x,y)$ and the string tension points inward,
\[\boxed{T=-2\lambda l=m\left(l\dot\theta^2+g\cos\theta\right)}.\]Rolling hoop on an inclined plane
Let $s$ be distance measured down a plane of angle $\alpha$, and choose the sense of $\theta$ so rolling without slipping is
\[f=s-R\theta=0.\]For a body of mass $M$, radius $R$, and centre-of-mass moment of inertia $I$,
\[L=\frac12M\dot s^2+\frac12I\dot\theta^2+Mgs\sin\alpha.\]The multiplier equations are
\[M\ddot s-Mg\sin\alpha=\lambda, \qquad I\ddot\theta=-\lambda R.\]Using $\ddot s=R\ddot\theta$ gives
\[\boxed{ \ddot s=\frac{g\sin\alpha}{1+I/(MR^2)}}.\]For a thin hoop, $I=MR^2$, so
\[\boxed{\ddot s=\frac12g\sin\alpha},\]with units $\mathrm{m\,s^{-2}}$. The multiplier is negative for the chosen $s$ direction, corresponding to static friction acting up the plane.
Principle of least action
Hamilton’s action $S=\int L\,dt$ is stationary for paths with fixed times and fixed configuration endpoints. At fixed energy, Maupertuis’ form of the principle is
\[\boxed{ \delta W=0, \qquad W=\int_{q_1}^{q_2}\sum_i p_i\,dq_i}.\]The name “least action” is traditional. The required value is stationary and need not always be a minimum.
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