Lagrange Equations of the Second Kind and Their Applications

The previous chapter derived Lagrange's equations. This one is about using them. "Equations of the second kind" is Lagrange's own name for the form that uses a minimal set of independent generalised coordinates and carries no constraint forces — the form $\dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q_j} - \dfrac{\partial L}{\partial q_j} = Q_j^{\text{nc}}$ we will apply throughout. (The "first kind" keeps all coordinates and enforces the constraints with multipliers; it appears here only in Sections 5 and 6, where the constraint force is the quantity we want.)

The programme is a tour of the systems every graduate course returns to: the central-force problem, small oscillations and normal modes, motion in a rotating frame, the symmetric top, and constrained systems handled by multipliers. In each case the work is the same four steps — choose coordinates, write $T$ and $V$, form $L$, differentiate — and the payoff is an equation of motion obtained without a single free-body diagram.

How to read this chapter. Treat each section as a worked template you will reuse. Section 3 (normal modes) and Section 8 (the symmetric top) are the two you are most likely to be examined on; Sections 5–6 (multipliers) are the tool for "find the tension" or "find where the bead leaves the track" problems. The recurring lesson is that a well-chosen coordinate makes a conserved quantity or a decoupling manifest before you differentiate anything.

1. The Recipe and the Role of Conserved Quantities

For a holonomic system:

1. Choose $n$ independent generalised coordinates $q_1,\dots,q_n$.
2. Write the kinetic energy $T(\vec q,\dot{\vec q},t)$ by substituting $\dot{\vec r}_i = \sum_j (\partial\vec r_i/\partial q_j)\dot q_j + \partial\vec r_i/\partial t$ into $\sum_i \tfrac12 m_i \dot{\vec r}_i^{\,2}$, and the potential $V(\vec q,t)$.
3. Form $L = T - V$ (add $-U$ for a velocity-dependent potential; keep non-conservative forces as $Q_j^{\text{nc}}$).
4. For each $j$, evaluate $\dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q_j} - \dfrac{\partial L}{\partial q_j} = Q_j^{\text{nc}}$.

Before integrating, look for structure. A cyclic coordinate $q_c$ (absent from $L$) gives $p_c = \partial L/\partial\dot q_c = \text{const}$ immediately. If $\partial L/\partial t = 0$, the Jacobi energy function $h = \sum_j \dot q_j\,\partial L/\partial\dot q_j - L$ is conserved; for a scleronomic system $h = T + V$. Each conserved quantity is a first integral that lowers the order of the problem by one.

2. The Central-Force Problem

A particle of mass $m$ in a central potential $V(r)$, working in the plane of the motion with polar coordinates $(r,\phi)$:

$\phi$ is cyclic, so $p_\phi = m r^2\dot\phi \equiv \ell$ is conserved — this is the angular momentum, and its constancy is Kepler's second law. The $r$-equation is

The two-dimensional problem has collapsed to one-dimensional motion in the effective potential $V_{\text{eff}}$, whose centrifugal term $\ell^2/2mr^2$ was generated automatically by $\partial T/\partial r$. Since $\partial L/\partial t = 0$, the energy $E = \tfrac12 m\dot r^2 + V_{\text{eff}}(r)$ is also conserved, and $\dot r = \pm\sqrt{(2/m)(E - V_{\text{eff}})}$ reduces the problem to a quadrature.

Worked example — Kepler. For $V = -k/r$, $V_{\text{eff}} = -k/r + \ell^2/2mr^2$ has a single minimum at $r_0 = \ell^2/mk$; circular orbits sit there, bound orbits oscillate in $r$ between turning points, and integrating $d\phi/dr = (\ell/r^2)/\sqrt{(2m)(E - V_{\text{eff}})}$ gives the conic $r = \dfrac{\ell^2/mk}{1 + e\cos\phi}$.

3. Small Oscillations and Normal Modes

Near a stable equilibrium $\vec q = \vec q_0$ (a minimum of $V$), expand to quadratic order. Let $\eta_j = q_j - q_{0j}$. Then

with $M_{jk} = M_{kj}$ the (constant) mass matrix evaluated at $\vec q_0$ and $K_{jk} = \partial^2 V/\partial q_j\partial q_k|_{\vec q_0}$ the stiffness matrix. Lagrange's equations become the linear system

Seeking solutions $\eta_k = a_k e^{i\omega t}$ turns this into the generalised eigenvalue problem

The $n$ roots $\omega_r^2$ are the normal-mode frequencies; the corresponding eigenvectors $\vec a^{(r)}$ are the mode shapes, and the general motion is their superposition. The modes are $M$-orthogonal, $\vec a^{(r)\mathsf{T}} M\,\vec a^{(s)} = 0$ for $r \ne s$, which is what lets the coupled system be rewritten as $n$ independent oscillators in normal coordinates $\xi_r$.

Worked example — two masses, three springs. With equal masses $m$ and equal springs $k$, $M = m\,\mathbb{1}$ and $K = \begin{pmatrix} 2k & -k \\ -k & 2k \end{pmatrix}$. The eigenvalues are $\omega^2 = k/m$ (mode $(1,1)$, the masses moving together, the middle spring unstretched) and $\omega^2 = 3k/m$ (mode $(1,-1)$, the masses moving oppositely).

4. Motion in a Rotating Frame from a Lagrangian

Let a frame rotate at constant angular velocity $\vec\Omega$ about a fixed axis. Writing the Lagrangian of a free particle in the inertial frame and substituting $\vec v_{\text{in}} = \vec v_{\text{rot}} + \vec\Omega\times\vec r$ gives

Applying Lagrange's equations in the rotating coordinates reproduces the equation of motion

with the Coriolis term coming from the middle (velocity-linear) piece of $L$ and the centrifugal term from the last piece. The rotating-frame fictitious forces are thus not postulated — they fall out of a coordinate change applied to the Lagrangian, exactly as the centrifugal term did for the central-force problem.

5. Lagrange Multipliers for Holonomic Constraints

Sometimes the constraint force is the answer. Keep the constraint $f(\vec q,t) = 0$ explicit and adjoin it with a multiplier $\lambda(t)$:

together with $f(\vec q,t) = 0$. This is $n+2$ equations (the $n+1$ Lagrange equations plus $f=0$) for the $n+2$ unknowns $q_1,\dots,q_{n+1},\lambda$ (one more coordinate is kept than strictly independent). The term $\lambda\,\partial f/\partial q_j$ is the generalised force of constraint, so once $\lambda(t)$ is known the reaction is known.

Worked example — particle on a sphere, when does it leave? A particle slides on the outside of a fixed sphere of radius $R$ under gravity, starting near the top. Use spherical coordinates with $f = r - R = 0$. The $r$-equation gives $m(\ddot r - r\dot\theta^2) = -mg\cos\theta + \lambda$, and on the constraint $\ddot r = 0$, $\dot r = 0$, so $\lambda = m(g\cos\theta - R\dot\theta^2)$. Energy conservation gives $R\dot\theta^2 = 2g(1 - \cos\theta)$, hence $\lambda = mg(3\cos\theta - 2)$. The normal force $\lambda$ reaches zero at $\cos\theta = 2/3$: the particle leaves the sphere there.

6. Non-Holonomic Constraints

For a constraint that is a non-integrable relation among the velocities,

the same multiplier structure applies, now with one multiplier $\lambda_\alpha$ per constraint:

These $n$ equations plus the $m$ constraint relations determine the $n$ coordinates and $m$ multipliers. The method works because the virtual displacements still satisfy $\sum_j a_{\alpha j}\,\delta q_j = 0$ (frozen time), so the constraint forces $\sum_\alpha \lambda_\alpha a_{\alpha j}$ do no virtual work — the same principle as always.

Worked example — vertical disc rolling on a plane. Coordinates $(x,y,\theta,\phi)$: contact-point position, heading, spin. The rolling constraints $\dot x = a\dot\phi\cos\theta$, $\dot y = a\dot\phi\sin\theta$ are non-holonomic. To get the friction force out of the multipliers, the constraint must not be substituted into $L$ beforehand: write $L = \tfrac12 m(\dot x^2 + \dot y^2) + \tfrac12 I_3\dot\phi^2 + \tfrac12 I_1\dot\theta^2$ (disc on a horizontal plane, no potential) for the four coordinates treated as independent, and adjoin the two rolling constraints with multipliers $\lambda_1,\lambda_2$. The $x$- and $y$-equations then read $m\ddot x = \lambda_1$, $m\ddot y = \lambda_2$: the multipliers are the horizontal friction-force components at the contact point. For steady rolling ($\dot\theta,\dot\phi = \text{const}$) these combine into a centripetal force of magnitude $ma\dot\phi\dot\theta$ directed toward the centre of the path — the disc rolls in a circle of fixed radius, with friction supplying exactly the force needed to curve it. (Substituting the constraint into $L$ first, as if to simplify, removes all $\dot x,\dot y$ dependence and makes the multipliers vanish identically — it answers a different question, the motion of the already-constrained system, not the force of constraint.)

7. Systems with a Velocity-Dependent Potential

The charged particle is the standard case. With $L = \tfrac12 m\dot{\vec r}^{\,2} - e\phi + e\,\dot{\vec r}\cdot\vec A$, in a uniform magnetic field $\vec B = B\hat z$ and the symmetric gauge $\vec A = \tfrac12 B(-y,x,0)$, cylindrical coordinates $(\rho,\varphi,z)$ give

$\varphi$ and $z$ are cyclic. The $\varphi$-momentum $p_\varphi = m\rho^2\dot\varphi + \tfrac12 e B\rho^2$ is conserved (canonical angular momentum), and $p_z = m\dot z$ is conserved. The $\rho$-equation then yields circular cyclotron motion at $\omega_c = eB/m$ drifting uniformly along $z$ — the helical path, obtained without ever writing $\vec v\times\vec B$.

8. The Symmetric Top

A rigid body with $I_1 = I_2 \ne I_3$, one point fixed, in gravity. Using Euler angles $(\phi,\theta,\psi)$ — precession, nutation, spin — the kinetic energy is

and with the centre of mass a distance $\ell$ from the fixed point, $V = M g\ell\cos\theta$. Both $\phi$ and $\psi$ are cyclic, giving two conserved momenta:

With $\partial L/\partial t = 0$ the energy is conserved as well. Eliminating $\dot\phi$ and $\dot\psi$ leaves a one-dimensional problem for $\theta$ in an effective potential $V_{\text{eff}}(\theta)$, whose analysis gives steady precession, nutation between two values of $\theta$, and the "sleeping top" stability condition $I_3^2\omega_3^2 > 4 I_1 M g\ell$.

9. Further Worked Systems

Bead on a rotating hoop (a bifurcation). A bead slides on a circular hoop of radius $R$ spun about its vertical diameter at rate $\omega$. With $\theta$ the angle from the bottom, $L = \tfrac12 m R^2\dot\theta^2 + \tfrac12 m R^2\omega^2\sin^2\theta + m g R\cos\theta$. The equation of motion is $R\ddot\theta = (R\omega^2\cos\theta - g)\sin\theta$. For $\omega^2 < g/R$ the only stable equilibrium is $\theta = 0$; for $\omega^2 > g/R$ that point becomes unstable and a new stable equilibrium appears at $\cos\theta_* = g/R\omega^2$ — a pitchfork bifurcation produced entirely by the $T_0$ term of the rotating constraint.

Double pendulum, linearised. For equal masses $m$ and equal rod lengths $\ell$, expanding the Lagrangian of Chapter 2 to quadratic order in $(\theta_1,\theta_2)$ gives $M = m\ell^2\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, $K = mg\ell\begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$, and $\det(K - \omega^2 M) = 0$ yields $\omega^2 = (2\pm\sqrt2)\,g/\ell$: two normal modes, a slow in-phase swing and a fast out-of-phase one.

Common Pitfalls

• Using the conserved $p_\phi$ to eliminate $\dot\phi$ inside $L$ before differentiating. Substituting a first integral back into the Lagrangian and then varying gives the wrong equations (a Routhian must be used instead). Derive all equations of motion from the full $L$, then impose the conserved quantity.
• Forgetting the centrifugal term in $V_{\text{eff}}$. It comes from $\partial T/\partial r$, not from the potential; a common slip is to write $V_{\text{eff}} = V(r)$ and lose the barrier that prevents collapse to $r = 0$.
• Wrong count of multiplier equations. With the multiplier method you keep more coordinates than degrees of freedom; the extra equations are the constraint relations themselves.
• Assuming the disc's rolling constraint is holonomic. On a straight line it is; free to steer on a plane it is not, and only the multiplier method (Section 6) applies.
• Linearising $T$ but not evaluating $M$ at equilibrium. The mass matrix must be the constant $M_{jk}(\vec q_0)$; keeping its $\vec q$-dependence spoils the constant-coefficient structure and the normal-mode analysis.
• Mixing up the Euler angles. $\phi$ is precession about the vertical, $\psi$ is spin about the symmetry axis, $\theta$ is the tilt; the kinetic energy depends on which convention you fixed.

Summary

• Lagrange's equations "of the second kind" use independent generalised coordinates and carry no constraint forces; apply the four-step recipe and then exploit cyclic coordinates ($p_c$ conserved) and time-translation ($h$ conserved).
• Central force: $\phi$ cyclic gives $\ell = m r^2\dot\phi$; the radial motion is one-dimensional in $V_{\text{eff}}(r) = V(r) + \ell^2/2mr^2$, reducing the orbit to a quadrature (the conic for $V = -k/r$).
• Small oscillations: expand about a stable equilibrium to get $M\ddot{\vec\eta} + K\vec\eta = 0$; the normal-mode frequencies solve $\det(K - \omega^2 M) = 0$ and the $M$-orthogonal eigenvectors decouple the motion.
• Rotating frame: substituting $\vec v_{\text{in}} = \vec v_{\text{rot}} + \vec\Omega\times\vec r$ into $L$ produces the Coriolis and centrifugal forces automatically.
• Multipliers (first kind): $\dfrac{d}{dt}\dfrac{\partial L}{\partial\dot q_j} - \dfrac{\partial L}{\partial q_j} = \sum_\alpha\lambda_\alpha a_{\alpha j}$ keeps the constraints explicit; $\lambda_\alpha\,a_{\alpha j}$ is the constraint force, used to find tensions, normal forces and separation points, and to handle non-holonomic (rolling) systems.
• Symmetric top: $\phi,\psi$ cyclic give $p_\phi,p_\psi$; with energy conservation the motion reduces to one dimension in $\theta$, yielding precession, nutation and the sleeping-top condition $I_3^2\omega_3^2 > 4I_1 Mg\ell$.