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A smooth wire has instantaneous shape $z=f(r)$ in cylindrical coordinates and rotates at constant angular velocity $\dot{\theta }=\omega$. The tangent vector at fixed time is ${\mathbf{e}}_r+f^{\prime }(r){\mathbf{e}}_z$. Its dot product with the constraint force vanishes. Projecting Newton's second law along this tangent vector gives the equation in the title.

Multiplying that equation by $\dot{r}$ proves conservation ofThis is a conserved quantity in the rotating description; the motor can exchange ordinary mechanical energy with the bead. A relative equilibrium at $a>0$ satisfies $U^{\prime }(a)=g{f}^{\prime }(a)-{\omega }^{2}a=0$. Its linearized equation is $(1+f^{\prime }(a{)}^2)\ddot{ϵ}=-U^{\prime \prime }(a)ϵ$. A strict local minimum with $U^{\prime \prime }(a)>0$ gives Lyapunov stability, while $U^{\prime \prime }(a)<0$ gives an unstable equilibrium. If $U^{\prime \prime }(a)=0$, higher orders are needed.

Constraint power is the rate of actual work done by constraint forces. For particles with Cartesian velocities ${\mathbf{v}}_i$ and constraint forces ${\mathbf{F}}_i^{\mathrm{c}}$, the total isIt enters the balance of kinetic energy and, after accounting for conservative applied forces, mechanical energy. An ideal mechanical constraint instead imposes zero total virtual work, ${\sum }_i{\mathbf{F}}_i^{\mathrm{c}}\cdot \delta {\mathbf{r}}_i=0$, for allowed virtual displacements at fixed time. Those displacements need not be parallel to actual velocities when the permitted geometry moves.^[ref]

For a smooth ideal restriction $f(\mathbf{r},t)=0$ on one particle, write ${\mathbf{F}}^{\mathrm{c}}=\lambda \nabla f$. Differentiation along its motion gives $\nabla f\cdot \mathbf{v}+{\partial }_{t}f=0$, henceA fixed smooth surface has ${\partial }_{t}f=0$ and zero constraint power. A time-dependent holonomic constraint can have nonzero constraint power even without friction. For example, a bead carried upwards at constant speed $u$ on a horizontal smooth wire receives normal force $mg$ and power $mgu$, increasing its gravitational potential energy. The drive moving the wire supplies that energy.

Zero total constraint power also does not require zero power on every body. Consider two hanging masses connected by a taut, massless, inextensible string over an ideal massless, frictionless pulley. The shared tension is $T$, and the two endpoint speed magnitudes are $v$. If one mass rises while the other falls, the string supplies powers $+Tv$ and $-Tv$: it transfers energy between the bodies while supplying zero net energy to the pair.

With $m_1=0.20\mathrm{kg}$, $m_2=0.30\mathrm{kg}$ and $g=9.81m/{\mathrm{s}}^2$, Newton's second law givesAt $v=0.50m/s$, the two tension powers are $+1.1772\mathrm{W}$ and $-1.1772\mathrm{W}$. Their cancellation leaves gravity's net power, $(m_2-m_1)gv=0.4905\mathrm{W}$, equal to $dK/dt=(m_1+m_2)av$. This constraint power transfer distinguishes an ideal connection that redistributes energy from an externally driven moving support that adds energy to the chosen system.

Constraint power transfer is redistribution of energy between parts of a mechanically connected system through constraint forces. An ideal connection may exert nonzero power on each body while exerting zero total power on the combination. Power on a body is the actual force–velocity scalar product; the zero-virtual work property of an ideal mechanical constraint concerns the sum over admissible virtual displacements.^[ref]

An original ideal rigid link calculation makes the cancellation explicit. Let two endpoint positions be ${\mathbf{r}}_A$ and ${\mathbf{r}}_B$, joined by a massless link of fixed length $\ell$. SetFor axial tension $T$, the link exerts $+T\mathbf{n}$ on A and $-T\mathbf{n}$ on B. The powers supplied to the endpoints areThe endpoint velocities need not be identical: the link can rotate. Only their components along the link must agree.

For stipulated $T=12\mathrm{N}$ and common axial endpoint speed $0.40m/s$, the link supplies $+4.8\mathrm{W}$ to A and $-4.8\mathrm{W}$ to B. It transfers energy without supplying net energy to their combined system. A load cell measuring axial force and tracked endpoint positions measuring velocities would permit these two powers to be compared. External forces determine how the endpoints subsequently accelerate.

Equal and opposite forces alone do not prove power cancellation. If the connector stretches, $\dot{\ell }=\mathbf{n}\cdot ({\mathbf{v}}_B-{\mathbf{v}}_A)$ and the same axial-force calculation givesAn elastic connector can store the corresponding elastic energy; a dissipative connector can convert mechanical energy into heat. Likewise, an externally prescribed moving support can deliver nonzero net constraint power to a subsystem. The location of the system boundary determines whether a measured exchange is internal transfer or externally supplied work.

An ideal mechanical constraint is a restriction whose constraint forces do zero total virtual work for every admissible virtual displacement at a fixed time:The condition concerns the permitted variations of the whole chosen system. It does not require every constraint force to vanish, and it does not necessarily imply zero actual constraint power. Stewart's treatment distinguishes a displacement along a moving constraint from a virtual displacement made while its time is held fixed.^[ref]

For smooth independent holonomic constraints $f_a({\mathbf{r}}_1,\dots ,{\mathbf{r}}_N,t)=0$, allowed variations obeyConstraint forces of the form ${\mathbf{F}}_i^{\mathrm{c}}={\sum }_a{\lambda }_a{\nabla }_i{f}_a$ therefore satisfy the ideal condition, with Lagrange multipliers ${\lambda }_a$. The argument explains why reaction forces disappear from the virtual-work equation without being absent from the apparatus.

For an original stationary example, constrain a particle of mass $0.20\mathrm{kg}$ to a fixed horizontal circle of radius $0.50\mathrm{m}$. At speed $3m/s$ the required radial force magnitude is $m{v}^2/R=3.6\mathrm{N}$. The allowed virtual displacement is tangential, so its scalar product with that radial normal reaction is zero. The force nevertheless produces the centripetal acceleration needed to maintain the trajectory.

A moving support shows the limitation of equating ideality with energy conservation. For the Cartesian constraint $x=a(t)$, admissible virtual displacements satisfy $\delta x=0$, whereas the actual velocity satisfies $\dot{x}=\dot{a}$. A reaction $F_x$ has zero virtual work but actual power $F_x\dot{a}$. An actuator moving the support can thus supply or remove mechanical energy. A force readout combined with the support's measured speed would detect that exchange.

For a fixed holonomic restriction, actual velocities obey the same homogeneous tangency condition as virtual displacements, giving zero total constraint power. For linked bodies, that total can cancel while individual powers are nonzero, as in constraint power transfer. Friction and compliance require their own model: calling a surface a mechanical constraint does not establish the ideal virtual-work property.

An ideal rigid link constrains the distance between two attachment points to a fixed length while transmitting an axial constraint force. In the model used here the link is massless, its end joints rotate freely, and it carries tension or compression without bending or stretching. For endpoint positions ${\mathbf{r}}_A,{\mathbf{r}}_B$,This is a holonomic constraint, with reactions that can be represented by Lagrange multipliers. The virtual-work treatment of constrained motion distinguishes these reaction forces from the applied forces that remain in reduced equations.^[ref]

Define the axial unit vector $\mathbf{n}=({\mathbf{r}}_B-{\mathbf{r}}_A)/\ell$. Differentiating the fixed-length condition givesThus endpoint velocities have equal axial components, while transverse relative motion can rotate the link. For an original planar example, let ${\mathbf{r}}_B-{\mathbf{r}}_A=(0.30,0.40)\mathrm{m}$, so $\ell =0.50\mathrm{m}$ and $\mathbf{n}=(0.6,0.8)$. The relative velocity $(-0.8,0.6)m/s$ is perpendicular to the link and corresponds to angular speed $2rad/s$. The length stays constant even though the endpoint velocities differ.

A second derivative exposes the required relative acceleration:In the example it is $-2m/{\mathrm{s}}^2$ along the link. Preserving length does not permit setting the relative acceleration to zero during rotation.

For axial force $+T\mathbf{n}$ on A and $-T\mathbf{n}$ on B, their total actual power is $T\mathbf{n}\cdot ({\mathbf{v}}_A-{\mathbf{v}}_B)=0$. Individual powers can be nonzero, giving constraint power transfer. A freely hinged rigid strut can carry either sign of axial force in this idealization; a taut ideal string can transmit tension but cannot enforce fixed separation by compression when it goes slack.

A real linkage approximates the model only while axial extension, joint friction and link inertia are negligible for the chosen motion. Tracking endpoint separation during loading tests the assumed length constraint; a load cell and position measurements can test the associated force and power balance.

In cylindrical coordinates,Differentiating the position vector $\mathbf{r}=r{\mathbf{e}}_r+z{\mathbf{e}}_z$ twice gives the velocity and accelerationThis derives acceleration in cylindrical coordinates rather than assuming it.

At a fixed time, the wire has parametrization $r\mapsto r{\mathbf{e}}_r+f(r){\mathbf{e}}_z$. Its derivative is the nonzero tangent vectorA smooth wire exerts a constraint force $\mathbf{R}$ with $\mathbf{R}\cdot \mathbf{t}=0$. Because $g$ is the gravitational acceleration, the gravitational force is $-mg{\mathbf{e}}_z$; the printed $-g{\mathbf{e}}_z$ is interpreted per unit mass. Projecting Newton's second law, $m\ddot{\mathbf{r}}=\mathbf{R}-mg{\mathbf{e}}_z$, along $\mathbf{t}$ eliminates $\mathbf{R}$:The chain rule gives $\ddot{z}=f^{\prime }(r)\ddot{r}+f^{\prime \prime }(r){\dot{r}}^2$. Rearrangement yieldsThis is the governing equation for a bead on a uniformly rotating wire. Its azimuthal acceleration contributes nothing to the projection because ${\mathbf{e}}_{\theta }\cdot \mathbf{t}=0$.

In cylindrical coordinates, $\theta =\omega t$ and $z=r^2/(2a)$. ThusThe acceleration isA tangent vector along the wire is $\mathbf{t}={\mathbf{e}}_r+(r/a){\mathbf{e}}_z$. Because the wire is smooth, its constraint force is orthogonal to $\mathbf{t}$. Taking the dot product of Newton's second law with $\mathbf{t}$ eliminates that force; gravitational weight contributes $-mgr/a$. HenceMultiplying by $a^2/m$ gives

In vacuum, the source-free Maxwell equations arePut $\psi =\omega (t-z/c)$ and denote the constant electric amplitude by $E_0$. A corresponding traveling-wave magnetic field isBoth divergences vanish because both the electric field and the magnetic field depend only on $z,t$ and have zero $z$ component. Direct differentiation gives $\nabla \times \mathbf{E}=(\omega E_0/c)(\cos \psi ,\sin \psi ,0)=-{\partial }_t\mathbf{B}$ and $\nabla \times \mathbf{B}=(\omega E_0/c^2)(-\sin \psi ,\cos \psi ,0)=c^{-2}{\partial }_t\mathbf{E}$. A time-independent background magnetic field satisfying the homogeneous equations could be added; the formula is the wave component.

Introduce the magnetic vector potential and electric potential by $\mathbf{B}=\nabla \times \mathbf{A}$ and $\mathbf{E}=-\nabla \varphi -{\partial }_t\mathbf{A}$. These identities automatically give the divergence equation for $\mathbf{B}$ and Faraday's law. For this wave choose $\varphi =0$ and $\mathbf{A}=(c/\omega )\mathbf{B}$; the derivatives just calculated verify both potential equations.

The electromagnetic force is the Lorentz force, $Q(\mathbf{E}+\mathbf{v}\times \mathbf{B})$. On the smooth constraint plane $z=z_0$, the constraint force is normal to the plane. Both $\mathbf{v}$ and $\mathbf{B}$ are tangential, so $\mathbf{v}\times \mathbf{B}$ is normal and is balanced by that constraint force; it does not affect the tangential motion. ThusA particular motion is ${\mathbf{r}}_{\parallel }={\mathbf{r}}_0-[Q{E}_0/(m{\omega }^2)](\cos \psi ,\sin \psi )$, a circle of radiusThe general planar motion also includes a constant drift velocity; the particle moves on a circle when that drift is zero. The constraint force can do no work because it is perpendicular to the velocity.