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Applied Differential Equations with Insightmaker and Data (Fall 2025 Ed.)

Erich McAlister

Section 3.5 Conserved Quantities

Definition 3.5.1. Consevred Quanity (two-dimentional).

Let \(x_1,\ldots,x_n\) be the dependent variables of a system of \(n\) differential equations. A conserved quantity for this system is a scalar-valued function \(C(x_1,\ldots,x_n)\) such that \(C(x_1(t),\ldots,x_n(t))\) is a constant function of \(t\text{.}\)
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Example 3.5.2.

We have encountered one conserved quantity in Subsection 3.3.1. In that model, the quantity \(S(t) + I(t) + R(t)\) remained constant. This was illustrated using the areas graph in the simulation.
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Conserved quatities often occur as sums of stocks. The conservation law takes graphical form in the following diagram:
Figure 3.5.3. The basic structure for the quantity \(x_1+x_2+\cdots +x_n\) to be conserved.
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The most important conserved quantity in many physical situations is energy. We explore the case of the undamped and unforced harmonic oscillator below.
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Example 3.5.4. The Undamped and Unforced Harmonic Oscillator is Conservative.

\begin{equation*} mx'' + kx =0 \end{equation*}
\(x' = v\)
\begin{align*} x' =\amp\ v \\ v' =\amp\ -\frac{k}{m}x \end{align*}
\begin{equation*} E(x,v) = \frac{k}{2}x^2 + \frac{m}{2}v^2. \end{equation*}
\(x\)\(v\text{.}\)\(t\)
\begin{equation*} \frac{d}{dt}(E(x(t),v(t))) = k x x' + m v v' = kxv - kxv = 0. \end{equation*}
\(KE' = mvv'\text{.}\)
Figure 3.5.5. A harmonic oscillator with computed energies.
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A generalization of the above for a second order system occurs when the system has the form
\begin{equation*} \vec{X}'' = -\nabla U(\vec{X}). \end{equation*}
In this case, \(U\) is called the potnetial function. The total energy is then given by
\begin{equation*} E(\vec{X}) = \frac{1}{2}m\|\vec{X'}\|^2 + u(\vec{X}). \end{equation*}
The reader is encouraged to try modeling the motion of a particle in the plane under the influence of various potential functions. It’s quite interesting to see what you get!
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Energy arguments are unbiquitous in physics. In the next section, we present a scenario in which energy transfer can be used to decide when a toy car with a skidding wheel stops accelerating.
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