Lagrange's Equations of the Second Kind

Lagrange's equations of the second kind provide a systematic way to derive equations of motion using independent generalized coordinates without explicitly introducing reactions of ideal constraints.

Generalized coordinates

If a system has $s$ degrees of freedom, its configuration can be described by independent coordinates $q_1,\ldots,q_s$. These may be linear displacements, angles, or other parameters that uniquely determine configuration.

Generalized velocities

The derivatives $\dot q_j$ are generalized velocities. The kinetic energy is written as a function $T(q_j,\dot q_j,t)$.

Generalized forces

The virtual work of active forces is written:

$$\delta A=\sum_{j=1}^{s}Q_j\delta q_j,$$

where $Q_j$ is the generalized force corresponding to coordinate $q_j$.

Lagrange's equations

For a system with ideal constraints:

$$\frac{d}{dt}\left(\frac{\partial T}{\partial\dot q_j}\right)-\frac{\partial T}{\partial q_j}=Q_j,\qquad j=1,\ldots,s.$$

The number of independent equations equals the number of degrees of freedom.

Conservative forces

If forces have potential energy $\Pi(q,t)$, their conservative generalized-force contribution is $Q_j=-\partial\Pi/\partial q_j$. With the Lagrangian $L=T-\Pi$, the equations may be written $d(\partial L/\partial\dot q_j)/dt-\partial L/\partial q_j=Q_j^{nc}$, where $Q_j^{nc}$ are nonconservative generalized forces.

Solution procedure

1. determine the number of degrees of freedom;
2. choose independent $q_j$;
3. express positions and velocities through $q_j,\dot q_j$;
4. calculate $T$;
5. determine $Q_j$ or $\Pi$;
6. write one Lagrange equation for each coordinate.

Example

For a mass $m$ on a horizontal spring with coordinate $x$, $T=m\dot x^2/2$ and $\Pi=kx^2/2$. Lagrange's equation gives $m\ddot x+kx=0$.

Common mistakes

• choosing dependent coordinates as independent;
• assuming every generalized force has units of newtons — for an angular coordinate it has units of moment;
• omitting coordinate dependence of kinetic energy;
• counting the same conservative force both through $\Pi$ and through $Q_j$.