mathematics//dynamical systems//balance equation
How much there is, how fast it changes and how much accumulates are three different quantities tied together by a balance.
How much there is, how fast it changes and how much accumulates are three different quantities tied together by a balance.
A time derivative is an instantaneous rate of change. If V(t)V(t)V(t) is the water volume in a tank, V˙(t)\dot V(t)V˙(t) is its rate: volume per time, not accumulated volume and not volume divided by the clock reading.
V˙(t)=limΔt→0V(t+Δt)−V(t)Δt.\dot V(t)=\lim_{\Delta t\to0}\frac{V(t+\Delta t)-V(t)}{\Delta t}.V˙(t)=Δt→0limΔtV(t+Δt)−V(t).
The integral accumulates rates. Knowing the initial volume and the net flow,
V(t)=V(t0)+∫t0tV˙(τ) dτ.V(t)=V(t_0)+\int_{t_0}^{t}\dot V(\tau)\,d\tau.V(t)=V(t0)+∫t0tV˙(τ)dτ.
τ\tauτ runs over the interior instants of the integral; here it is neither a delay nor a time constant.
The balance builds the equation. For a tank with no internal generation, V˙=qin−qout\dot V=q_{\mathrm{in}}-q_{\mathrm{out}}V˙=qin−qout expresses conservation. Closing the model still needs an outflow law, for example qout=khq_{\mathrm{out}}=khqout=kh within an approximately linear regime. With V=ahV=ahV=ah and constant section aaa,
ah˙=qin−kh.a\dot h=q_{\mathrm{in}}-kh.ah˙=qin−kh.
The section is not a conductance: it converts flow into level velocity. That distinction is the nodal capacity of a physical network.
The general recipe behind every such equation: the derivative of a quantity equals what makes it increase minus what makes it decrease, and the mechanisms are added one term at a time, an interaction term, a feedback, an external forcing, a spatial diffusion (differential equations).
Signs live at different levels. h>0h>0h>0 is a positive level; h˙<0\dot h<0h˙<0 says it is falling; h¨>0\ddot h>0h¨>0 says its velocity is increasing, perhaps becoming less negative. A positive second derivative does not guarantee the level is already rising. For a position qqq, q˙\dot qq˙ is velocity, q¨\ddot qq¨ acceleration and q(3)q^{(3)}q(3) jerk: successive rates with different units. More derivatives does not mean better control.
An integral relation is not an explicit solution.
h(t)=h(0)+∫0tqin(τ)−kh(τ)a dτh(t)=h(0)+\int_0^t \frac{q_{\mathrm{in}}(\tau)-kh(\tau)}{a}\,d\tauh(t)=h(0)+∫0taqin(τ)−kh(τ)dτ
is the same dynamics written another way, with the unknown function still inside the integral. Writing it this way has not solved it.
A proportional loss leads to exponential decay. Several coupled rates define a vector field (phase portrait). An algorithm that accumulates rates in finite steps is a numerical scheme (discretization), not a new physical law.