# Particle-Spring Systems

• Fear always springs from ignorance. (Ralph Waldo Emerson, American Poet, Lecturer and Essayist, 1803-1882)

Let $x(t)$ be the position at time $t$ of a unit point mass or particle moving without friction along a line subject to a linear spring force $F(x) = -x$,  see Intro to Springs.

Newton’s equations of motion take the form:

• $dx = vdt, \quad dv = -xdt$.
Particle-spring system: One particle/mass gliding without friction along a line attached to one of a spring attached to a fixed wall: Here $u(t)=x(t)$ is the position at time $t$ measured from some the reference point with zero spring force:

# Watch

• How does a spring function?
• How to motivate that spring force is proportional to elongation?

# Conservation of Total Energy

The total energy $E$ a particle connected to a linear spring modeled by $\dot x=v$ and $\dot v=-x$ is defined by

• $E=\frac{1}{2}(x^2+v^2)$.

Let us now prove that the Trapezoidal Method conserves the total energy. This follows by multiplying the time-stepping equations

• $x^{n+1}-x^n=\frac{1}{2}(v^{n+1}+v^n)dt, \quad v^{n+1}-v^n=-\frac{1}{2}(x^{n+1}+x^n)dt$

by $\frac{1}{2}(x^{n+1}+x^n)$ and $\frac{1}{2}(v^{n+1}+v^n)$, respectively, to get by summation and reordering (using that $(a+b)(a-b)=a^2-b^2$),

• $E^{n+1}\equiv\frac{1}{2}((x^{n+1})^2+(v^{n+1})^2)=\frac{1}{2}((x^{n})^2+(v^{n})^2)\equiv E^n$

which expresses conservation of the total energy as $E^{n+1}=E^{n}$.

We understand that as the particle moves back and forth, kinetic energy is transformed into elastic energy stored as the spring stretches or compresses, which is transformed back into kinetic energy as the stretching and compression is eased.