Generalized Forces and Generalized Momenta

The manipulator equation of modern robotics is equation (7) wearing industrial clothing: $M(q)\ddot q + C(q,\dot q)\dot q + g(q) = \tau$, where $M(q)$ is exactly the mass matrix of Derivation B, $g(q)$ collects the gravitational generalized forces of the worked example, and the motor torques $\tau$ are the non-conservative $Q_j^{\mathrm{nc}}$. Every physics engine that trains today's humanoid robots and quadrupeds — MuJoCo, Drake, Isaac — assembles and inverts this equation millions of times per training run, in joint coordinates, because that is the space where constraints vanish and motors live. When a 2026-era humanoid catches a thrown package, a computer has just solved for generalized forces along sixty-odd coordinates in under a millisecond. The concept's reach goes far beyond machines. Thermodynamics organizes itself around the same conjugacy: pressure is the generalized force conjugate to volume, surface tension to area, chemical potential to particle number — "whatever multiplies $\delta q$ in the work" is one of physics' most reused sentences. And the canonical momentum $\mathbf{p} = m\mathbf{v} + q\mathbf{A}$ is the reason your MRI scanner's physics and your quantum mechanics course agree: the operator $-i\hbar\nabla$ represents the canonical momentum, so magnetic couplings enter quantum mechanics exactly as Lagrange's derivative said they must.

Configuration space is a surface of allowed shapes: each point is one arrangement of the whole system, and moving along the $q_j$-direction deforms the system in one specific, constraint-respecting way. The real forces $\mathbf{F}_i$ live out in ordinary 3D space and point wherever they point. The generalized force $Q_j$ is the answer to a sharper question: of all that force, how much actually pushes along the $q_j$-direction of allowed motion? Formally it is a dot product with $\partial\mathbf{r}_i/\partial q_j$; informally it is the shadow the physical forces cast on the system's freedom.

The projection picture explains at a stroke the two most useful facts about $Q_j$. First, constraint forces vanish from the bookkeeping: a frictionless wire pushes only perpendicular to itself, and a perpendicular force dotted with an allowed displacement gives zero. You never compute the normal force, the string tension, or the hinge reaction — they are filtered out at the definition. Second, units follow the coordinate: if $q_j$ is an angle, $\partial\mathbf{r}_i/\partial q_j$ has the length of a lever arm, and $Q_j$ comes out as a torque. Work is the universal currency; $Q_j$ is denominated in whatever makes $Q_j\,\delta q_j$ come out in joules.

Generalized momentum answers the mirror-image question: how does the dynamics grip the velocity $\dot q_j$? For a free particle in Cartesian coordinates the answer is boring, $p_x = m\dot x$. But the definition $p_j = \partial L/\partial\dot q_j$ takes orders from the Lagrangian, not from Newton. Curve the coordinates and $p_j$ becomes a mixture of several velocities weighted by a configuration-dependent mass matrix. Add a magnetic field and it silently absorbs a piece of the field itself, $q\mathbf{A}$. The single most instructive mistake in this chapter — one you should make once, carefully, and never again — is to write $p_j = m\dot q_j$ on autopilot. The pairing is symmetric and worth savoring: $Q_j$ comes from differentiating work with respect to a coordinate; $p_j$ from differentiating $L$ with respect to a velocity. The Euler–Lagrange equation is the bridge that balances their rates, and because every object in sight is built from derivatives of scalars, the whole structure survives any smooth change of coordinates untouched.

Start with the definition, which is nothing more than a change of basis applied to work. Displace the system virtually — freeze time, wiggle the configuration by $\delta q_j$ in a way the constraints allow — and tally the work done by the applied forces. Written once in particle language and once in coordinate language, that tally defines the $Q_j$:

where: $\mathbf{F}_i$ — the net applied (non-constraint) force on particle $i$; $\delta\mathbf{r}_i$ — its virtual displacement; $\delta q_j$ — the independent virtual changes of the coordinates. Because the $\delta q_j$ can be varied one at a time, the coefficients $Q_j$ are uniquely determined — the definition is not circular, it is a component extraction. The machinery of virtual displacements itself belongs to the principle of virtual work and d'Alembert's principle; here we harvest it. Unpacking $\delta\mathbf{r}_i = \sum_j (\partial\mathbf{r}_i/\partial q_j)\,\delta q_j$ by the chain rule and swapping the two sums turns the definition into the formula you actually compute with:

where: $\partial\mathbf{r}_i/\partial q_j$ — the vector telling how particle $i$ moves per unit change of $q_j$, with every other coordinate and time held fixed. Read it as a projection: each particle's force is dotted onto the direction that coordinate actually drags that particle. For the bead of the hero diagram, $\mathbf{F} = (0,-mg)$ and $\partial\mathbf{r}/\partial x = (1, f'(x))$ give $Q_x = -mg\,f'(x)$ — gravity matters only through the wire's slope, and the normal force never even enters. When the applied forces come from a potential $V(q_1,\dots,q_n)$, the projection in (2) is exactly a gradient component, and the generalized force collapses to

where: $V$ — the potential energy expressed directly in the generalized coordinates. This is the form that gets absorbed into $L = T - V$: conservative generalized forces stop being a separate object and become part of the Lagrangian's landscape. Anything not derivable from $V$ — friction, drag, a motor drive — stays outside and re-enters below as an explicit right-hand side.

Now the mirror object. Instead of asking how work responds to a coordinate, ask how the Lagrangian responds to a velocity:

where: $L(q,\dot q,t) = T - V$ — the Lagrangian; $p_j$ — the momentum canonically conjugate to $q_j$. The three names in circulation — generalized, conjugate, canonical — all denote this one derivative. Its units, like those of $Q_j$, follow the coordinate: $p_j\,\dot q_j$ always carries the units of energy, so an angular coordinate breeds an angular momentum. For the vast standard class of mechanical systems, $T$ is quadratic in the velocities, $T = \tfrac12\sum_{kl} M_{kl}(q)\,\dot q_k\dot q_l$, and $V$ is velocity-independent. Differentiating and using the symmetry $M_{kl} = M_{lk}$ gives the momentum in its most physical form:

where: $M_{jk}$ — the configuration-dependent mass (inertia) matrix. Two lessons sit in this formula. The momentum conjugate to one coordinate is in general a blend of all the velocities — in a two-link robot arm, $p_{\theta_1}$ contains $\dot\theta_2$, because swinging the forearm moves mass that the shoulder angle also carries. And even in one dimension the effective inertia can depend on position: the bead on the parabolic wire below has $p_x = m\,(1 + f'^2)\,\dot x$, larger where the wire is steep because a unit of $\dot x$ then implies extra vertical speed. With both conjugate objects defined, the Euler–Lagrange equation — derived from stationary action in its own article — can be read as a physical sentence rather than a formula:

where: $\partial L/\partial q_j$ — the Lagrangian's own generalized force, containing both $-\partial V/\partial q_j$ and any coordinate-dependence of the kinetic energy (the origin of centrifugal-type terms). This is Newton's second law in coordinate-free dress: rate of change of conjugate momentum equals conjugate force. Its most celebrated corollary costs nothing: if $q_j$ is absent from $L$, the right side vanishes and $p_j$ is conserved — the cyclic-coordinate mechanism that the next article develops into a systematic tool. Real machines have friction and drives. Forces with no potential simply stay in the projected form (2) and sit on the right-hand side:

where: $\mathbf{F}_i^{\mathrm{nc}}$ — the non-conservative forces (friction, air drag, motor torques). The conservative part of the physics lives inside $L$; everything else is projected and appended. This split is exactly how robot dynamics engines organize the manipulator equation, with motor torques appearing as the $Q_j^{\mathrm{nc}}$.

Which formula answers which question

Everything so far assumed $V$ carries no velocities. Electromagnetism breaks that assumption in the most consequential way in physics. The Lagrangian that reproduces the Lorentz force on a charge $q$ in potentials $\phi,\ \mathbf{A}$ is

where: $\phi$ — scalar potential; $\mathbf{A}$ — vector potential, $\mathbf{B} = \nabla\times\mathbf{A}$. The velocity-linear coupling $q\,\dot{\mathbf{r}}\cdot\mathbf{A}$ is a velocity-dependent potential — a genuinely new structural feature, and the reason the momentum conjugate to position acquires a field contribution.

1 · Bead on a parabolic wire (confidence builder)

A bead of mass $m = 0.2\ \mathrm{kg}$ slides on a frictionless parabolic wire $y = x^2/2R$ with $R = 0.5\ \mathrm{m}$, in gravity. Using $x$ as the generalized coordinate, find the generalized force $Q_x$ and the conjugate momentum $p_x$ at the moment the bead passes $x = 0.3\ \mathrm{m}$ with $\dot x = 1.5\ \mathrm{m/s}$.

Step 1. Geometry first. The position is $\mathbf{r} = (x,\ x^2/2R)$, so $\partial\mathbf{r}/\partial x = (1,\ x/R)$ with slope $f'(x) = x/R = 0.3/0.5 = 0.6$ at the given point.

Step 2. Generalized force by projection, eq. (2): with $\mathbf{F} = (0,\ -mg)$,

The sign says the force pushes toward decreasing $x$ — downhill. The normal force never entered: it is perpendicular to $\partial\mathbf{r}/\partial x$ by construction.

Step 3. Momentum from the Lagrangian. The speed obeys $v^2 = \dot x^2 + \dot y^2 = (1 + f'^2)\,\dot x^2$, so $L = \tfrac12 m (1 + f'^2)\,\dot x^2 - mg\,x^2/2R$ and

Step 4. Compare with the naive answer: $m\dot x = 0.300\ \mathrm{kg\,m/s}$ — a 36% error. The effective inertia $m(1+f'^2)$ is the one-coordinate mass matrix $M_{xx}(x)$: where the wire is steep, a unit of $\dot x$ drags extra vertical speed, and the momentum knows it. No field, no exotic units — geometry alone already separates $p_x$ from $m\dot x$.

2 · Holding torque for a two-link robot arm (exam level)

A planar two-link arm has link 1: $m_1 = 3.0\ \mathrm{kg}$, length $l_1 = 0.40\ \mathrm{m}$, centre of mass at $l_{c1} = 0.20\ \mathrm{m}$; link 2: $m_2 = 1.5\ \mathrm{kg}$, centre of mass at $l_{c2} = 0.15\ \mathrm{m}$ from the elbow. Joint angles are measured from the horizontal: $\theta_1 = 30°$ at the shoulder, and the elbow adds $\theta_2 = 45°$, so link 2 sits at $\theta_1 + \theta_2 = 75°$. Find the gravitational generalized forces $Q_{\theta_1}, Q_{\theta_2}$ and the motor torques needed to hold the pose — first bare, then carrying a $2.0\ \mathrm{kg}$ payload at the wrist, $l_2 = 0.30\ \mathrm{m}$ from the elbow.

Step 1. Set up the potential in the joint coordinates. Heights above the shoulder: $y_{c1} = l_{c1}\sin\theta_1$ for link 1's centre of mass, $y_{c2} = l_1\sin\theta_1 + l_{c2}\sin(\theta_1+\theta_2)$ for link 2's. So $V = g\big[(m_1 l_{c1} + m_2 l_1)\sin\theta_1 + m_2 l_{c2}\sin(\theta_1+\theta_2)\big]$.

Step 2. Generalized forces by eq. (3), $Q_j = -\partial V/\partial\theta_j$ — note both angles move link 2, so $\theta_1$ collects both terms:

Step 3. Numbers. $m_1 l_{c1} + m_2 l_1 = 3.0\times0.20 + 1.5\times0.40 = 1.20\ \mathrm{kg\,m}$; $m_2 l_{c2} = 0.225\ \mathrm{kg\,m}$; $\cos 30° = 0.866$, $\cos 75° = 0.259$. Then $Q_{\theta_1} = -9.81\,(1.20\times0.866 + 0.225\times0.259) = -(10.19 + 0.57) = \boxed{-10.8\ \mathrm{N\,m}}$ and $Q_{\theta_2} = -9.81\times0.225\times0.259 = \boxed{-0.571\ \mathrm{N\,m}}$. These are torques, not forces — the coordinate is an angle.

Step 4. Static balance via eq. (7): holding still means $\dot p_j = 0$ and $\partial L/\partial\theta_j$ reduces to $Q_{\theta_j}^{\mathrm{grav}}$, so the motors must supply $\tau_j = -Q_{\theta_j}$: $\tau_1 = +10.8\ \mathrm{N\,m}$, $\tau_2 = +0.571\ \mathrm{N\,m}$ — the shoulder works nineteen times harder than the elbow, because it carries every gram at full lever arm.

Step 5. Add the payload: it hangs at height $l_1\sin\theta_1 + l_2\sin(\theta_1+\theta_2)$, so append $m_p l_1\cos\theta_1$ to the $\theta_1$ bracket and $m_p l_2\cos(\theta_1+\theta_2)$ to both. Recomputing: $Q_{\theta_1} = -10.8 - 9.81\,(2.0\times0.40\times0.866 + 2.0\times0.30\times0.259) = \boxed{-19.1\ \mathrm{N\,m}}$ and $Q_{\theta_2} = -0.571 - 9.81\times2.0\times0.30\times0.259 = \boxed{-2.09\ \mathrm{N\,m}}$. A 2 kg cup of coffee nearly doubles the shoulder torque. This calculation — equation (11) evaluated at every control tick — is called gravity compensation, and it runs inside every collaborative robot on every factory floor.

References. Goldstein, Poole & Safko, Classical Mechanics, 3rd ed., §1.4, §2.6 · Landau & Lifshitz, Mechanics, 3rd ed., §§7–8 · Morin, Introduction to Classical Mechanics, Ch. 6 · Taylor, Classical Mechanics, §7.8

Provenance. Definitions and the d'Alembert-to-Euler-Lagrange derivation follow Goldstein, Poole & Safko §1.4 and Landau–Lifshitz Mechanics §§7–8. A verifier should confirm: (1) the chain-rule derivation of $Q_j=\sum_i\mathbf F_i\cdot\partial\mathbf r_i/\partial q_j$; (2) the 'dot-cancellation' identities used to convert the inertial term into $T$-derivatives; (3) the EM Lagrangian yields the Lorentz force and $\mathbf p=m\dot{\mathbf r}+q\mathbf A$ (Goldstein §1.5); (4) the central-force result $p_\phi=mr^2\dot\phi$ conserved. All numerical/structural relations are standard textbook results, no fabricated constants. Worked-example numbers on this page ($Q_x = -1.18\ \mathrm{N}$ and $p_x = 0.408\ \mathrm{kg\,m/s}$ for the bead; $\tau_1 = 10.8\to19.1\ \mathrm{N\,m}$, $\tau_2 = 0.571\to2.09\ \mathrm{N\,m}$ for the arm) are illustrative arithmetic recomputed from equations (2), (3), (5), (11).

Builds on: Generalized Coordinates and Configuration Space · Virtual Displacements and the Principle of Virtual Work · d'Alembert's Principle · The Lagrangian Function

Leads to: The Euler-Lagrange Equations · Cyclic Coordinates and Conservation of Conjugate Momenta · The Energy Function and Jacobi Integral · Symmetries and Continuous Transformations