Mechanical energy balance for a potential presentation: $E^{\mathrm{\prime }}={\mathrm{\partial }}_{t}U$ plus the residual power, conservation for a static potential with no residual forces, and the integrated work form

Adopt the force model, one inertial frame, and the potential presentation of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (a): a $C^2$ potential $U:I_0\times Q⟶\mathbb{R}$ of dimension $\mu {\mathrm{\ell }}^2{\tau }^{-2}$ and residual forces $d_i$ of the standing regularity, with $F_i(t,x,v)=-{\mathrm{\nabla }}_{x_i}U(t,x)+d_i(t,x,v)(i=1,\dots ,N)$ on the state domain. Let $x:J\to Q$ be a motion of the model on an interval $J$ with nonempty interior, write ${\mathfrak{d}}_i(t):=d_i(t,x(t),w(t))$ for the evaluated residual forces and $E(t):=K(t)+U(t,x(t))$ for the mechanical energy of the definition (clause (b)). Then:

1. $E$ is differentiable on $J$ and $E^{\mathrm{\prime }}(t)={\mathrm{\partial }}_{t}U(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩(t\in J):$ the mechanical energy changes at the rate of the explicit time dependence of the potential plus the power of the residual forces;
2. if ${\mathrm{\partial }}_{t}U\equiv 0$ on $I_0\times Q$ and each evaluated residual force ${\mathfrak{d}}_i$ vanishes identically along this motion, then $E$ is constant on $J$. If, more strongly, every residual-force map $d_i$ vanishes on its entire state domain, the model is a global potential model with the declared representative $U$;
3. for all $t_0<t_1$ in $J$, $K(t_1)-K(t_0)=U(t_0,x(t_0))-U(t_1,x(t_1))+{\int }_{t_0}^{t_1}\left[{\mathrm{\partial }}_{t}U\right(t,x(t))+{\sum }_i⟨{\mathfrak{d}}_i(t),w_i(t)⟩]dt,$ equivalently $W([t_0,t_1])=U(t_0,x(t_0))-U(t_1,x(t_1))+{\int }_{t_0}^{t_1}[{\mathrm{\partial }}_{t}U+{\sum }_i⟨{\mathfrak{d}}_i,w_i⟩]dt$: the work of the total force is the decrease of the potential energy plus the work of the explicit time dependence and of the residual forces.

The representative $U$ is the one declared by the model; replacing $U$ by $U+c(t)$ changes $E$ by $c(t)$ and $E^{\mathrm{\prime }}$ by $c^{\mathrm{\prime }}(t)$, as recorded in Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets clause (c). The statement is read in the fixed frame; no empirical content is claimed.

• Three distinct contributions. The balance separates the explicit time dependence of the potential (${\mathrm{\partial }}_{t}U$), the power of the residual forces, and the potential difference; a model with a time-dependent potential or with residual forces is driven, and its mechanical energy is not conserved in general. The companion worked example computes all three terms on explicit data.

• Conservation is conditional and representation-dependent. Clause 2 requires the representative $U$ to be time-independent and the residual forces to vanish; a potential $U+c(t)$ with time-dependent $c$ turns a conserved model into a presentation with $E^{\mathrm{\prime }}=c^{\mathrm{\prime }}(t)$, as clause (c) of Potential presentations of a force model, potential and mechanical energy, closed versus exact force fields, local potentials and equipotential level sets records. Energy statements therefore name the representative.

• Frame dependence of conservation. Under the scalar transfer of Transformation of the state quantities under a change of inertial frame: centre of mass, total momentum, kinetic energy, angular momentum, torque, power, work and mechanical energy clause 5, with its boost velocity $v$, $\tilde{E}=E+⟨v,RP⟩+\frac{1}{2}M\parallel v{\parallel }_2^2$. Thus if both $E$ and $P$ are constant, then $\tilde{E}$ is constant. If $P$ varies, constancy of $E$ alone does not imply constancy of $\tilde{E}$; the transferred potential can also have explicit time dependence.