Chapter 10 Reduced and alternative variational principles

10.1 Routhian procedure for cyclic variables

In a system with \(n\) Lagrange coordinates \(q^\alpha \), suppose that \(q^n\) is a cyclic (ignorable, kinosthenic) variable, and let us denote by \(q^{\bar \alpha }\) the first \(n-1\) Lagrange coordinates. Noether’s implies that \(p_n= \frac {\partial L}{\partial \dot q^n}=c_n\) is a constant of the motion. Suppose that this relation allows to express \(\dot q^n\) as a function of the other variables

\begin{equation} \label {eq:2aa} \dot q^n=U^n(q^{\bar \alpha },\dot q^{\bar \alpha },c_n,t). \end{equation}

Note that this implies that

\begin{equation} \label {eq:467} \frac {\partial L}{\partial \dot q^n}\big |_{\dot q^n=U^n}=c_n. \end{equation}

It follows that the Lagrangian equations of motion, i.e., a system of \(n\) differential equations of second order can be reduced to a system of \(n-1\) differential equations of second order.

The Routhian procedure provides a reduced Lagrangian \(\bar L(q^{\bar \alpha },\dot q^{\bar \alpha },t)\) such that the reduced system are the Euler-Lagrange equations of the reduced Lagrangian. For the starting point variational principle, we have

\begin{equation} \label {eq:67} \delta \int ^{t_f}_{t_i}dt\, L=\int ^{t_f}_{t_i}dt\, \frac {\delta L}{\delta q^\alpha }\delta q^\alpha +{\big [\frac {\partial L}{\partial \dot q^\alpha }\delta q^\alpha \big ]}^{t_f}_{t_i} \end{equation}

On a natural trajectory, this reduces to

\begin{equation} \label {eq:417} \delta \int ^{t_f}_{t_i}dt\, L\approx {\big [p_{\bar \alpha }\delta q^{\bar \alpha }\big ]}^{t_f}_{t_i} +c_n{\big [\delta q^n\big ]}^{t_f}_{t_i}, \end{equation}

Since the last term can be written as \(\delta \int _{t_i}^{t_f}c_n\dot q^1\), this suggest that the appropriate reduced action principle, that gives rise to the reduced system of equations for arbitrary variations \(\delta q^{\bar \alpha }\) that vanish at the end points is

\begin{equation} \label {eq:464} \boxed {\bar L(q^{\bar \alpha },\dot q^{\bar \alpha },t)=(L-p_n\dot q^n)|_{\dot q^n=U^n} =L|_{\dot q^n=U^n}-c_n U^n}. \end{equation}

It is applicable for trajectories that have the same constant \(p_n\) than the natural trajectory. Indeed, the reduced system of equations is

\begin{equation} \label {eq:466} \big [\frac {d}{dt}\frac {\partial L}{\partial \dot q^{\bar \alpha }}\big ] \big |_{\dot q^n=U^n,\ddot q^n=\frac {d}{dt}U^n} =\frac {\partial L}{\partial \dot q^{\bar \alpha }}\big |_{\dot q^n=U^n}, \quad \frac {\partial L}{\partial \dot q^n}=c_n, \end{equation}

while the Euler-Lagrange equations for the reduced Lagrangian are

\begin{equation} \label {eq:465} -\frac {\delta \bar L}{\delta q^{\bar \alpha }}=\frac {d}{dt} \big [\frac {\partial L}{\partial \dot q^{\bar \alpha }}\big |_{\dot q^n=U^n} +\cancel {\frac {\partial L}{\partial \dot q^{n}}\big |_{\dot q^n=U^n}\frac {\partial U^n}{\partial \dot q^{\bar \alpha }}} -\cancel {c_n\frac {\partial U^n}{\partial \dot q^{\bar \alpha }}}\big ]- \frac {\partial L}{\partial q^{\bar \alpha }}\big |_{\dot q^n=U^n} -\cancel {\frac {\partial L}{\partial \dot q^{n}}\big |_{\dot q^n=U^n}\frac {\partial U^n}{\partial q^{\bar \alpha }}} +\cancel {c_n \frac {\partial U^n}{\partial q^{\bar \alpha }}}=0, \end{equation}

which indeed agree when using equation (10.2).

Finally, note that one may also arrive to the reduced action principle by using a Lagrange multiplier,

\begin{equation} \label {eq:478} L^\lambda (q^{\bar \alpha },\dot q^{\bar \alpha },\dot q^n,t,\lambda )=L-p_n\dot q^n-\lambda (p_n-c_n). \end{equation}

10.2 Parametrized systems

The solution of the Euler-Lagrange equations of motion may be given in parametrized form. Rather than using \(q^\alpha =q^\alpha (t)\), one introduces a parameter \(\tau \) and describes the solution in parametrized form through \(t=t(\tau ), q^\alpha =q^\alpha (\tau )\). If \(t' =\frac {d t}{d\tau }\neq 0\), \(q^{\prime \alpha }=\frac {d q^\alpha }{d\tau }\), the appropriate variation principle involves as independent variable \(\tau \) and contains \(n+1\) Lagrange coordinates \(q^\alpha ,t\). It is given by

\begin{equation} \label {eq:2} S=\int _{\tau _i}^{\tau _f} d\tau \, L^P,\quad L^P(q^\alpha ,t,q^{\prime \alpha },t^\prime )=t^\prime L(q^\alpha ,\dot q^\alpha ,t) |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}. \end{equation}

Indeed,

\begin{equation} \label {eq:468} \begin{split} &-\frac {\delta L^P}{\delta q^\alpha }=\frac {d}{d\tau }[\cancel {t^\prime }\frac {\partial L}{\partial \dot q^\alpha } _{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}\cancel {\frac {1}{t^\prime }}]-t^\prime \frac {\partial L}{\partial q^\alpha }|_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }},\\ &-\frac {\delta L^P}{\delta t}=\frac {d}{d\tau }(L|_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }} -\frac {\partial L}{\partial \dot q^\alpha }\big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}\frac {q^{\prime \alpha }}{t^\prime }) -t^\prime \frac {\partial L}{\partial t}\big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}. \end {split} \end{equation}

When using that

\begin{equation} \label {eq:469} \frac {d^2}{dt^2}q^\alpha =\frac {d}{dt}(\frac {q^{\prime \alpha }}{t^\prime }) =\frac {1}{t^\prime }\frac {d}{d\tau }(\frac {q^{\prime \alpha }}{t^\prime }) =\frac {q^{\prime \prime \alpha }}{(t^\prime )^2}-\frac {q^{\prime \alpha }t^{\prime \prime }}{(t^{\prime })^3}, \end{equation}

the first \(n\) Euler-Lagrange equations associated to (10.10) become

\begin{equation} \label {eq:470} t^\prime \big [\frac {\delta L}{\delta q^\alpha }\big ]\big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }, \ddot q^\alpha =\frac {q^{\prime \prime \alpha }}{(t^\prime )^2}-\frac {q^{\prime \alpha }t^{\prime \prime }}{(t^{\prime })^3}}=0, \end{equation}

and are thus equivalent to the original Euler-Lagrange equations of motion. This is the same reasoning as in the previous discussion concerning a change of the independent variable.

When working out the derivative with respect to \(\tau \) and using the previous discussion, one finds for the last expression of (10.10),

\begin{equation} \label {eq:471} -\frac {\delta L^P}{\delta t}=-q^{\prime \alpha }\big [\frac {\delta L}{\delta q^\alpha }\big ] \big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }, \ddot q^\alpha =\frac {q^{\prime \prime \alpha }}{(t^\prime )^2}-\frac {q^{\prime \alpha }t^{\prime \prime }}{(t^{\prime })^3}} \end{equation}

which vanishes as a consequence of the previous \(n\) equations.

Note that \(L^P\) does not define a regular system. Indeed, when denoting the \(n+1\) Lagrange coordinates \((t,q^\alpha )\) by \(q^A\), \(L^P(q^A,q^{\prime A})\) is homogeneous of degree \(1\) in the generalized velocities \(q^{\prime A}\),

\begin{equation} \label {eq:472} L^P(q^A,\mu q^{\prime A})=\mu L^P(q^A,q^{\prime A}), \end{equation}

which implies, by differentiation with respect to \(\mu \) and setting \(\mu =0\) (Euler’s theorem for homogeneous functions) that

\begin{equation} \label {eq:473} q^{\prime B}\frac {\partial L^P}{\partial q^{\prime B}}=L^P. \end{equation}

Another derivation with respect to \(q^{\prime A}\) yields

\begin{equation} \label {eq:474} \frac {\partial ^2 L^P}{\partial q^{\prime A}\partial q^{\prime B}}q^{\prime B}+\cancel {\frac {\partial L^P}{\partial q^{\prime B}}} =\cancel {\frac {\partial L^P}{\partial q^{\prime B}}}, \end{equation}

which means that \(q^{\prime B}\) is an eigenvector with zero eigenvalue of the matrix \(\frac {\partial ^2 L^P}{\partial q^{\prime A}\partial q^{\prime B}}\), so that this matrix is degenerate.

10.3 Maupertius’/ Jacobi’s principle

If \(L\) does not depend explicitly on time \(t\), \(t\) is a cyclic variable of the variational principle defined by \(L^P\). The associated conserved momentum, \(p_t\) is minus the energy of the original variational principle defined by \(L\),

\begin{equation} \label {eq:475} p_t=\frac {\partial L^P}{\partial t^\prime } = (L-p_\alpha \dot q^\alpha )\big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }} =-E\big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}=-\mathbf E, \end{equation}

for some constant \(\mathbf E\). If the original Lagrangian was regular, one may solve this equation for \(t^\prime =T^\prime (q^\alpha ,q^{\prime \alpha })\) provided that not all generalized velocities vanish. Indeed,

\begin{equation} \label {eq:477} \frac {\partial ^2 L^P}{(\partial t^\prime )^2}=(\cancel {\frac {\partial L}{\partial \dot q^\alpha }} -\cancel {\frac {\partial L}{\partial \dot q^\alpha }} -\dot q^\alpha \frac {\partial ^2 L}{\partial \dot q^\alpha \partial \dot q^\beta }) \big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}(-\frac {q^{\prime \beta }}{(t^\prime )^2})= \frac {1}{t^\prime } (\dot q^\alpha \frac {\partial ^2 L}{\partial \dot q^\alpha \partial \dot q^\beta }\dot q^\beta ) \big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}. \end{equation}

In this case, the variational principle of Maupertius/Jacobi is obtained by applying the Routhian procedure to \(L^P\). It gives rise to the reduced variational principle, applicable for trajectories with the same energy \(\mathbf E\) than the natural trajectory,

\begin{equation} \label {eq:476} \bar L^P=T^\prime L|_{\dot q^\alpha =\frac {q^{\prime \alpha }}{T^\prime }}+E|_{\dot q^\alpha =\frac {q^{\prime \alpha }}{T^\prime }}T^\prime =p_\alpha \big |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{T^\prime }}q^{\prime \alpha }. \end{equation}

when taking into account (10.17).

For an explicitly time-independent \(L\) of the form (9.13), one finds from (10.17) that

\begin{equation} \label {eq:482} \frac 12 g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }(T^{\prime })^{-2}+V=\mathbf E\Longrightarrow T^{\prime }=\pm \sqrt {\frac {g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }}{2(\mathbf E-V)}}, \end{equation}

where the sign follows that of \(t^\prime \). This implies on the one hand that

\begin{equation} \label {eq:484} t-t_i=\pm \int ^{\tau }_{\tau _i}d\tau ^\prime \sqrt {\frac {g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }}{2(\mathbf E-V)}} =\pm \int _\gamma \frac {\overline {ds}}{\sqrt {2(\mathbf E-V)}}, \end{equation}

where \(\gamma \) is a path from some initial configuration \(q^\alpha _i\) to a final confiuration \(q^\alpha \), and on the other that

\begin{equation} \label {eq:481} \bar L^P=\frac {1}{T^\prime }g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }+A_\alpha q^{\prime \alpha } =\pm \sigma \sqrt {|2(\mathbf E-V)|}\sqrt {|g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }|}+A_\alpha q^{\prime \alpha }, \end{equation}

where \(\sigma \) denotes the sign of \(g_{\alpha \beta }q^{\prime \alpha }q^{\prime \beta }\) and

\begin{equation} \label {eq:483} \boxed {\bar S^P=\int \big [\pm \sigma \sqrt {|2(\mathbf E-V)|}\overline {ds}+A_\alpha dq^{\alpha }\big ]}. \end{equation}

The Lagrange multiplier version of Maupertius’/Jacobi’s variational principle is

\begin{equation} \label {eq:488} S^{P,\lambda }=\int d\tau \,\big [p_\alpha |_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}q^{\prime \alpha } +\lambda [E|_{\dot q^\alpha =\frac {q^{\prime \alpha }}{t^\prime }}-\mathbf E]\big ]. \end{equation}

Remarks:

(i) In particular, if \(V=0=A_\alpha \), the variational principle of Maupertius/Jacobi for \(S=\int dt\, \frac {1}{2}g_{\alpha \beta }\dot q^\alpha \dot q^\beta \) reduces to the reparametrization invariant variational principle associated to \(\bar S^P=\pm \sigma \sqrt {2|\mathbf E|} \int \overline {ds}\).

(ii) In this context, a systematic derivation of the solution to the brachystochrone 5.2.1 starting from Hamilton’s principle is possible. The appropriate action is

\begin{equation} \label {eq:485} S[x,z]=\int _{t_*}^{t_0} dt\, \big [\frac 12 m (\dot x^2+\dot z^2)-mgz\big ], \end{equation}

with initial and final conditions \(x(t_*)=x^*,z(t_*)=z^*,x(t_0)=0,z(t_0)=0\). The objective is to determine the spatial path \(x(z)\) such that the time of descent is minimal. In order to use conservation of energy, the use of Maupertius’/Jacobi’s principle is appropriate. The initial conditions imply that \(\mathbf E=mgz^*\), so that equation (10.21) yields for the time of descent between \((x^*,z^*)\) and \((0,0)\)

\begin{equation} \label {eq:489} t_0-t_*=-\int _{z_*}^0dz\, \sqrt {\frac {1+(x^\prime )^2}{2g(z^*-z)}},\quad x^\prime =\frac {dx}{dz}, \end{equation}

in agreement with equation (5.48).

The remaining equations are the Euler-Lagrange equations associated to

\begin{equation} \label {eq:486} \bar S^P=-m\int dz \sqrt {2g(z^*-z)}\sqrt {1+(x^\prime )^2}. \end{equation}

Since \(x\) is a cyclic variable,

\begin{equation} \label {eq:487} p_x=\sqrt {2g(z^*-z)}\frac {x^\prime }{\sqrt {1+(x^\prime )^2}}=d, \end{equation}

for some constant \(d\geq 0\). The sign is fixed by requiring that \(x\) increases when \(z\) increases. Taking the square yields

\begin{equation} \label {eq:490} (x^\prime )^2=\frac {d^2}{2g(z^*-z)-d^2}=\frac {2R^\prime }{z^*-z-2R^\prime },\quad 2R^\prime =\frac {d^2}{2g} \end{equation}

Through integration, one then finds

\begin{equation} \label {eq:491} x-x^*=\int ^z_{z^*}dz' \frac {\sqrt {2R^\prime }}{\sqrt {z^*-z-2R^\prime }}=- \int ^{z-z^*}_{0}dZ \frac {\sqrt {2R^\prime }}{\sqrt {Z-2R^\prime }} \end{equation}

Setting \(Z=R^\prime (1-\cos \theta )=2R^\prime \sin ^2\frac {\theta }{2}\).

solve this to find \(x=x(z;x^*,z^*,c')\), and then compute the time of fall on-shell, then extremize it