Skip to main content

Applied Differential Equations with Insightmaker and Data (Fall 2024 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 \(t\text{.}\)

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.
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.
The most important conserved quantity in many physical situations is energy. We explore the case of the undamped and unforced harmonic oscillator below.

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.
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!
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.
PrevTopNext