Lagrangian Mechanics

More generally, a constraint is called anholonomic whenever it cannot be written in the form Equation (21.1); the inequality constraints of Definition 21.5 are one instance.

A constraint that does not depend explicitly on time is called scleronomous; one that does depend explicitly on time is called rheonomous [Goldstein:2002].

The hypothesis is not a technicality. Two constraint surfaces whose gradients become parallel at a point are, at that point, statically indeterminate: the same total constraint force can be apportioned between them in infinitely many ways, and no equation of motion can decide the split. The elementary instance is a table with four legs on a rigid floor, where the four normal forces are not determined by rigid-body mechanics and the real answer comes from the elasticity of Continuum Mechanics and Elasticity. Positive definiteness of Equation (21.24) is exactly the statement that this has not happened.

Since \(\vect{\Phi}^{s}_{k}=\lambda^{k}\nabla^{s}\phi^{k}\) is a force, the multiplier carries whatever unit makes the product a newton: \([\lambda^{k}]=\mathrm{N}\,\mathrm{m}/[\phi^{k}]\), equivalently \([\lambda^{k}\phi^{k}]=\mathrm{J}\), which is the mechanical face of the statement in Remark 16.42 that the augmented integrand must remain an energy. For the commonest case, a constraint written as a length — a rod of fixed length, a bead on a wire, a body resting on a surface — \(\phi^{k}\) is in metres and \(\lambda^{k}\) is a force in newtons outright. Writing the same constraint as \(\phi=r^{2}=R^{2}\) instead of \(\phi=r=R\) rescales the multiplier to \(\mathrm{N}/\mathrm{m}\) and halves it; the constraint force \(\lambda\nabla\phi\) is of course unchanged. The multiplier is not an invariant of the constraint, only of the constraint as written.

Proposition 21.25 is the promised instance of the distinctions drawn in Definition 21.5 and Definition 21.16. Resting on a surface is the inequality constraint \(r\geq R\), not the equality \(r=R\); the equality is merely the part of it that is active. At a point of the boundary \(r=R\) an outward virtual displacement is compatible with \(r\geq R\) while the inward one is not, so the compatible displacements there are not invertible, and the argument of Equation (21.14) — which required a displacement tangent to the surface — covers only the tangential ones. What survives is the inequality \(\vect{\Phi}^{s}\cdot\delta\vect{x}^{s}\geq0\): a one-sided constraint can do no negative virtual work. Its mechanical content is the sign condition \(\lambda\geq0\), and Equation (21.28) is the instant at which that condition ceases to hold. A model that keeps the equality past \(\theta_{\ast}\) predicts a surface that pulls the body back, which is not a small error but the wrong phenomenon.

The alternative treatment eliminates the constraint at the outset, taking \(\theta\) as the single generalized coordinate; the second-kind equations of Section 21.4.3 then reduce to Equation (21.30) with \(r=R\), namely \(R\ddot{\theta}=g\sin\theta\), and the normal force never appears. That is the right route when only the trajectory is wanted, and the wrong one whenever the quantity of interest is carried by the constraint: the tension in a suspension wire, the load on a bearing, the normal force that decides whether a rolling body slips, the reaction that a structure must be built to withstand. Those are the multipliers themselves, and the price of eliminating them is that they cannot be recovered afterwards from the reduced description.

The hypothesis \(\pp V/\pp\dot{q}^{a}=0\) can be weakened, and must be, because the Lorentz force on a charge is velocity-dependent and yet has a Lagrangian. Suppose the generalized force derives from a generalized potential \(U\left(q,\dot{q},t\right)\) in joules in the sense

Substituting Equation (21.42) into Equation (21.32) and collecting terms gives Equation (21.39) with \(\Lag=T-U\), by the same two lines as above; the extra total time derivative in Equation (21.42) is precisely what is needed to cancel the \(\pp U/\pp\dot{q}^{a}\) that now survives in \(\pp\Lag/\pp\dot{q}^{a}\). The electromagnetic case \(U=q_{\mathrm{e}}\left(\Phi-\dot{\vect{x}}\cdot\vect{A}\right)\), with \(q_{\mathrm{e}}\) the charge in coulombs, \(\Phi\) in volts and \(\vect{A}\) in \(\mathrm{T}\,\mathrm{m}\), belongs to Generalized Classical Field Theory; it is also the reason Theorem 21.46 below is stated for a potential \(V\left(q,\dot{q},t\right)\). Velocity-dependent forces that are not of the form Equation (21.42) — sliding friction is the standard example — have no Lagrangian and must be carried on the right-hand side of Equation (21.38) as they stand.

The name is opaque and the content is not. That \(\Lag\) does not depend on \(q^{a}\) says that the translation \(q^{a}\mapsto q^{a}+\varepsilon\), with the other coordinates held fixed, leaves the Lagrangian — and therefore the action — invariant. Theorem 21.36 is thus the simplest instance of Noether's first theorem (Theorem 16.84), and the conserved quantity it produces is the Noether charge of that one-parameter group. Two cases carry names: invariance under a rigid translation of the whole system conserves the total linear momentum, and invariance under a rigid rotation conserves the total angular momentum (Corollary 16.88). The reading in terms of symmetry is the one that survives into field theory and quantum mechanics, where no coordinate is “absent” in any literal sense.

It is not enough that \(T\) carry no explicit \(t\). A single particle in one dimension described from a frame translating at constant speed \(v\), \(x=q+vt\), has \(T=\tfrac12 m\left(\dot{q}+v\right)^{2}\) — no explicit time dependence whatever — and yet \(\dot{q}\,\pp T/\pp\dot{q}-T=\tfrac12 m\left(\dot{q}^{2}-v^{2}\right) \neq T\). What fails is homogeneity, not time-independence, and it is the transformation Equation (21.7) rather than \(T\) that must be free of \(t\). In the general rheonomous case Equation (21.34) gives \(T=T_{2}+T_{1}+T_{0}\), homogeneous of degrees two, one and zero in the \(\dot{q}^{a}\), and the same computation yields

which is a perfectly good constant of the motion when \(\pp\Lag/\pp t=0\) but is not the energy \(T+V\). The rotating frame of Rigid Bodies and Rotating Frames is the case that matters in practice: there \(h\) is the Jacobi constant of the restricted three-body problem, \(T_{0}\) is the centrifugal term, and confusing it with the energy is a standard way to get a wrong answer.

The general statement behind Equation (21.48) is Euler's theorem on homogeneous functions: if \(f\) is differentiable and \(f(\mu u)=\mu^{k}f(u)\) for all \(\mu>0\), then \(u^{i}\,\pp f/\pp u^{i}=k f(u)\). It is a result of multivariable calculus and belongs in Real Analysis, which does not at present state it. Nothing above rests on it: the kinetic energy is an explicit quadratic form, so Equation (21.48) is one line of differentiation, and Equation (21.49) is obtained the same way term by term. The general theorem is nevertheless wanted elsewhere in this treatise — the virial theorem of Central Forces and Statics and the thermodynamic Euler relation of Part VI — Classical Thermodynamics both use it in degrees other than two — and is recorded here as owed.

Equation (21.50) is a power balance in watts, and its two terms are physically distinct. The first, \(\dot{q}^{a}Q_{a}\), is the rate at which the non-conservative active forces do work on the system — the friction that drains a pendulum, the drive that sustains it. The second, \(-\pp\Lag/\pp t\), is the rate at which work is done by whatever imposes the explicit time dependence: the hand that shortens the string of a pendulum, the motor that spins a constraint. Neither is a failure of energy conservation; both record energy crossing the boundary of the system as it was drawn. Note also that the constraint forces appear in neither term, which is the whole economy of the Lagrangian method — they were eliminated at Equation (21.15) precisely because they do no virtual work.

\(\pp\Lag/\pp t=0\) says that the action is invariant under a shift of the origin of time, so Theorem 21.43 is Noether's theorem for time translation (Corollary 16.87), exactly as Theorem 21.36 is Noether's theorem for a translation in a coordinate. The two extra hypotheses — scleronomous constraints and a velocity-independent potential — are not needed for a conserved quantity; they are needed only for that quantity to be the energy, per Remark 21.39. It is worth being explicit about what is and is not assumed: nothing here derives conservation of energy from a deeper principle, since time-translation invariance is itself an empirical property of the interactions, tested wherever energy accounting is tested. What the theorem supplies is the exact link between the two.

The traditional name “principle of least action” overstates the result, and Equation (21.53) says only what is true: the first variation vanishes. For a mechanical Lagrangian the action is indeed a minimum over sufficiently short intervals, but past the first conjugate point — the kinetic focus — it is a saddle, never a maximum; for a harmonic oscillator of angular frequency \(\omega\) that happens after a time \(\pi/\omega\), half a period. The second-variation machinery that decides this is Section 16.4, and the statement for mechanics specifically is Remark 16.98. The physical reason the stationary path is the one that matters is not classical at all: in the path-integral formulation of Path-Integral Quantization every history contributes a phase \(\exp\left(\ii S/\hbar\right)\), and neighbouring histories interfere destructively except where \(S\) is stationary.

If the potential depends on the coordinates alone, \(V=V(q)\), the system is conservative: the hypotheses of Proposition 21.38 and, given scleronomous constraints, of Theorem 21.43 are met, so \(h=E\) and \(E\) is constant.

It is worth recording what has and has not been shown. d'Alembert's principle Equation (21.19) and Hamilton's principle Equation (21.53) both yield Equation (21.54) for a holonomically constrained system with forces derived from a potential, so on that class they are equivalent and the choice between them is one of convenience. They are not equivalent in general. d'Alembert's principle is the more robust of the two: it survives non-conservative forces, which merely appear as \(Q_{a}\) on the right-hand side, and it survives non-integrable constraints, which restrict \(\delta q^{a}\) without restricting the histories. Hamilton's principle needs a Lagrangian to exist, and needs the admissible histories to form a neighbourhood. The historical order is the reverse of the logical one: Hamilton's essays [Hamilton:1834] came ninety years after d'Alembert and half a century after [Lagrange:1788], and the variational principle is the later and narrower statement, not the foundation from which the rest descends [Goldstein:2002] [Landau:1976].

A holonomic constraint removes one configuration variable and, with it, one velocity: the counting Equation (21.5) is exact and the system genuinely lives on a lower-dimensional space. A nonholonomic constraint removes a velocity and no configuration at all. The vertical rolling disc of Example 16.48 has four configuration variables and two constraints Equation (16.44), leaving two velocity degrees of freedom — and yet every one of the four configurations is reachable, by rolling the disc round a suitable closed loop. There is therefore no reduced coordinate set to pass to, and Definition 21.9 must be read with care: the number of independent velocities and the dimension of the accessible set are different numbers, which for holonomic systems coincide. This is the mechanical content of non-integrability, and it is why the parking of a car is possible at all.

The last step is the one to hold on to. Given a configuration \(q^{a}\) and a velocity \(\dot{q}^{a}\) that satisfies Equation (21.55), the multipliers \(\mu_{\alpha}\) follow at once, and Equation (21.57) then fixes the accelerations. The motion of a nonholonomic system is therefore determined by its mechanical state alone, exactly as a Newtonian system is. This is not a formality; it is what the next subsection turns on.

The equations of a rolling body are the d'Alembert–Chetaev ones Equation (21.57). Three things settle it, in increasing order of force.

The physical realization of the constraint. Rolling without slipping is enforced by static friction at the contact point: a real force, exerted instant by instant, whose magnitude adjusts to whatever the state requires. It does no work, because the material point of the body at the contact is instantaneously at rest, which is precisely the Chetaev condition Equation (21.56). Nothing in the physical situation restricts which histories the body is permitted to consider, which is what Definition 21.55 assumes.

Determinism. By Remark 21.54 the d'Alembert–Chetaev system fixes the motion from \(\left(q,\dot{q}\right)\) alone. The vakonomic system does not: in Equation (21.59) the multipliers obey their own differential equations, so their initial values \(\lambda_{\alpha}(t_{0})\) are extra data, and one mechanical initial state gives rise to an \(m\)-parameter family of vakonomic motions. A disc released twice from the same place with the same velocity does the same thing twice; that observation alone excludes the vakonomic equations as a description of it, and it can be made in a lecture room.

The two answer different questions. The vakonomic equations are not wrong — they are the correct equations of the optimal-control problem in which the constraint is imposed on the admissible curves by decree and one asks which admissible curve extremizes the action. That is a real and useful problem, and the extra initial data \(\lambda_{\alpha}(t_{0})\) are what select a particular optimal trajectory in it. It is simply not the problem a rolling disc solves. The mathematical statement of the dichotomy, and the reason Theorem 16.41 does not cover the nonholonomic case, is Remark 16.49.

Two honest caveats. First, the standard mechanics texts [Goldstein:2002] [Landau:1976] present Equation (21.57) as the equations of a nonholonomic system, and the derivation above is theirs in substance; but this treatise's bibliography holds no reference reporting a controlled experiment that discriminates the two prescriptions on a rolling body, and the claim of observational adequacy above therefore rests on the accumulated agreement of nonholonomic rigid-body dynamics with measurement — the rolling and spinning experiments of Rigid Bodies and Rotating Frames — rather than on a single cited test. Second, systems whose constraints are not imposed from outside but follow from a degenerate Lagrangian are a third case entirely, governed by the Dirac–Bergmann algorithm of Constrained Hamiltonian Systems: the Dirac–Bergmann Formalism, and nothing here applies to them.

The conventions of Notation 21.1 continue to apply: \(a,b,\dots\) label the \(f\) generalized coordinates. They are not the frame indices of Section 13.1, which label directions in spacetime; where spacetime enters, in Section 21.8.1, it carries \(\mu,\nu,\dots\) as usual.

There is a one-line reason behind the computation. The action Equation (21.52) is a number attached to a path—a scalar, independent of how the path is labelled—and Hamilton's principle (Theorem 21.46) characterizes the motion as a stationary point of that number. Stationarity is a property of the functional, not of the chart used to write it, so the resulting equations cannot depend on the chart either. Theorem 21.63 is that observation made explicit, and Equation (21.68) records the precise sense in which it holds: not that the components \(\mathcal{E}_{a}\) are unchanged, but that they transform as the components of a covector, which is what makes the statement \(\mathcal{E}_{a}=0\) coordinate-independent.

A second, distinct freedom leaves the equations of motion untouched: adding to \(\Lag\) the total time derivative of any function \(F(q,t)\),

changes the action only by the boundary term \(F(q_{b},t_{b})-F(q_{a},t_{a})\), which is fixed once the endpoints are fixed and therefore does not affect the variation. The Lagrangian of a system is thus not unique; only its equivalence class modulo point transformations and total derivatives is physical.

That the equations are untouched can also be seen without mentioning the boundary at all, and this is the sharper statement: the Euler–Lagrange operator annihilates a total time derivative identically, on every curve, extremal or not. Writing \(G=\dd F(q,t)/\dd t=\left(\pp F/\pp q^{b}\right)\dot{q}^{b}+\pp F/\pp t\) for \(F\) of class \(C^{2}\), the velocity derivative of \(G\) is \(\pp G/\pp\dot{q}^{a}=\pp F/\pp q^{a}\), so

by the symmetry of mixed partial derivatives (Proposition 7.73). Since \(\mathcal{E}_{a}\) is linear in its argument, \(\mathcal{E}_{a}[\Lag']=\mathcal{E}_{a}[\Lag]\), which is the claim. A function with \(\mathcal{E}_{a}\equiv0\) is called a null Lagrangian; in one independent variable Equation (21.73) has a converse, every null Lagrangian being a total derivative, and it is the freedom Equation (21.72) that reappears as the gauge freedom of the electromagnetic potentials and as the boundary terms that must be added to the Einstein–Hilbert action.

A free system does not travel in straight lines through configuration space; it travels along the straightest available lines, the autoparallels of the metric its own kinetic energy defines. With a potential the right-hand side acquires \(-g^{cd}\pp_{d}V\), so \(V\) acts as a force pushing the motion off the geodesics. Every statement here is the general apparatus of Section 13.10 specialized to a Riemannian configuration space.

Example 21.69 is worth pausing on, because it isolates a distinction that the rest of this treatise depends upon. The Christoffel symbols Equation (21.77) are not zero, yet the plane is flat: inserting the symbols Equation (21.77) into the Riemann tensor Equation (13.306) gives \(R^{a}{}_{bcd}=0\) identically, as it must, since curvature is a tensor (Theorem 13.152) and vanishes in the Cartesian chart.

So in a flat space a nonvanishing connection means only one of two things, which are worth separating. Curvilinear coordinates in an inertial frame — Example 21.69 — produce \(\mathring{\Gamma}\) terms that are pure bookkeeping: no observer feels anything, and an accelerometer carried along reads zero. A non-inertial frame, obtained by a time-dependent point transformation, produces \(\mathring{\Gamma}\) terms that an accelerometer does register; these are the inertial (fictitious) forces of Definition 21.19 proper — centrifugal, Coriolis and their relatives. What the two cases share is that both are removable everywhere at once, by returning to inertial Cartesian coordinates, and they are removable precisely because \(R^{a}{}_{bcd}=0\). Curvature is what cannot be removed. Keeping these apart is what makes the equivalence principle a sharp statement rather than a slogan (The Equivalence Principle and Classical Tests).

With the mostly-plus signature this treatise uses, the interval itself is \(\dd s^{2}=g_{\mu\nu}\dd x^{\mu}\dd x^{\nu}\), negative on a timelike worldline, so \(\dd s\) is not the proper distance travelled by a massive particle and \(c^{2}\dd\tau^{2}=-\dd s^{2}\). Writing the action in terms of \(\dd\tau\) rather than \(\dd s\), as in Equation (21.79), keeps this consistent with the interval classification of Lorentz Transformations and with Example 21.74 below. A timelike geodesic maximizes proper time, so the leading minus sign is what makes \(S\) stationary at a minimum.

Every geometric object used above — manifold, chart, metric, connection, geodesic, curvature — is defined once and in full generality, for dimension \(D=p+q\) and signature \((p,q)\), in Differentiable Manifolds, Tensors, and Curvature. This chapter has developed no new mathematics; it has pointed out that the mathematics classical mechanics has used since Lagrange is the mathematics general relativity uses, and that what separates the theories is not the language but the physics it is asked to describe: in mechanics the geometry is a bookkeeping device for constraints imposed by hand, while in general relativity the geometry of spacetime is itself the dynamical variable, determined by its own field equations.