Reaction work and mechanical-energy balance for ideal holonomic constraints: vanishing power for scleronomic constraints and the rheonomic work term

Adopt a holonomic-constraint model of Holonomic-constraint models of the force branch: constraint data, the constrained equation, virtual displacements and the scleronomic/rheonomic distinction with datum $(O,k,G)$ and the ideal-reaction law of Adopted ideal-reaction law for holonomic constraints: the d'Alembert condition, its scope and its limits. Suppose, in addition to the model hypotheses, that $G$ is of class $C^3$ on $O$ and $D_{x}G(t,x)$ has rank $k$ for every $(t,x)\in O$, and that the force map is of class $C^1$ on $I_0\times Q\times ({\mathbb{R}}^3{)}^N$; these are hypotheses (h1)-(h2) of Ideal holonomic constraints determine the reaction: multiplier form, explicit acceleration, unique constrained motions and the reaction-power identity. Let $x:J\to Q$ be a constrained motion with ideal reaction $R$ on an interval $J\subseteq I_0$ with nonempty interior, with velocity $w=x^{\mathrm{\prime }}$ and multiplier $\lambda :J\to {\mathbb{R}}^k$ as in Ideal holonomic constraints determine the reaction: multiplier form, explicit acceleration, unique constrained motions and the reaction-power identity clause 1. Suppose further that the model's force map is presented through a potential with residual forces in the sense of clause (a) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets: a $C^2$ function $U:I_0\times Q\to \mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and maps $d_i:I_0\times Q\times ({\mathbb{R}}^3{)}^N\to {\mathbb{R}}^3$ with the standing regularity of clause (iii) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance, such that $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)(i=1,\dots ,N)$ on the state domain (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework for the dimensions). Write $K$ and $E(t):=K(t)+U(t,x(t))$ for the kinetic and mechanical energy of Total mass, total momentum, centre of mass, angular momentum, torque, kinetic energy, power and work of a particle model in an inertial frame clauses (b),(d) and of clause (b) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets, and ${\mathfrak{d}}_i(t):=d_i(t,x(t),w(t))$ for the evaluated residual forces. Then:

1. reaction power and reaction work. For every $t\in J$, $R(t)\cdot w(t)=-\lambda (t)\cdot G_t(t,x(t)),$ and for all $t_0<t_1$ in $J$, ${\int }_{t_0}^{t_1}R(t)\cdot w(t)dt=-{\int }_{t_0}^{t_1}\lambda (t)\cdot G_t(t,x(t))dt$. If the datum is scleronomic, then $R\cdot w\equiv 0$ on $J$ and the reaction performs no work over any subinterval.

2. mechanical-energy balance. $E$ is differentiable on $J$ and $E^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}U(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩-\lambda (t)\cdot G_t(t,x(t))(t\in J):$ the mechanical energy changes at the rate of the explicit time dependence of the potential, the power of the residual forces, and the constraint term $-\lambda \cdot G_t$, which vanishes identically for a scleronomic constraint.

3. conservation in the static scleronomic case. If the datum is scleronomic, ${\mathrm{\partial }}_{t}U\equiv 0$ on $I_0\times Q$, and ${\mathfrak{d}}_i\equiv 0$ for every $i$ along the motion, then $E$ is constant on $J$.

4. integrated work form. For all $t_0<t_1$ in $J$, $K(t_1)-K(t_0)=U(t_0,x(t_0))-U(t_1,x(t_1))+{\int }_{t_0}^{t_1}\left[{\mathrm{\partial }}_{t}U\right(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩-\lambda (t)\cdot G_t(t,x(t)\left)\right]dt\mathrm{.}$

The constraint term of clause 2 is not a force of the model's force map: it is the mechanical work performed by the ideal mechanism that maintains a time-dependent constraint, and clause 1 records that this work is exactly the virtual-work-free reaction's actual power. The statement is conditional on the adopted model, idealization and potential presentation; it asserts no measured value.

• Ideal does not mean workless. The vanishing of virtual work and the vanishing of actual work are different conditions, and for a moving (rheonomic) constraint the second fails in general: the mechanism's reaction can have a component along the actual velocity of the constrained body even though it annihilates every frozen virtual displacement. The companion bead-on-a-rotating-wire example computes a nonzero reaction power $2m{\mathrm{\Omega }}^{2}r{r}^{\mathrm{\prime }}$, while the companion pendulum check computes a vanishing one.

• Where the mechanical energy goes. For a rheonomic ideal constraint the term $-\lambda \cdot G_t$ is the power injected or removed by the constraint mechanism; it is not attributable to the model's residual forces and it is not a potential term. A reduced description in a moving chart or a reduced coordinate can have a conserved reduced energy that differs from the inertial mechanical energy, and that difference is the content of this term in those descriptions; the comparison is carried out by the companions and is not claimed here in general.

• Conservation is conditional and representation-dependent. Clause 3 gives sufficient conditions: a scleronomic constraint, a static potential and vanishing residual forces along the motion. They are not necessary; by clause 2 and the zero-derivative criterion, energy is constant exactly when its displayed total rate vanishes along the motion. For example, with one label, $O=\mathbb{R}\times {\mathbb{R}}^3$, $G(t,x)=x_1-vt$ for a constant nonzero speed $v$, $F=0$, $U=0$ and $x(t)=(vt,0,0)$, the regular constraint is rheonomic, but $R=0$, $\lambda =0$ and $E=m{v}^2\mathrm{/}2$ is constant. More generally the three rate terms can cancel. The normalization of $U$ by a time-dependent additive function changes $E$ and adds $c^{\mathrm{\prime }}(t)$ to the right-hand side of clause 2, exactly as in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c).