19.3 Second Order Differential Equations

A second order differential equation is one that expresses the second derivative of the dependent variable as a function of the variable and its first derivative. (More generally it is an equation involving that variable and its second derivative, and perhaps its first derivative.)

Perhaps the easiest way to handle such an equation is to give a name to the first derivative. Then the original equation becomes a pair of coupled equations for the dependent variable and for its derivative. What you get when doing this is a pair of first order differential equations like the pair of coupled equations seen in the Predator Prey problem.

Given the equation $$x'' = f(x,x',t)$$, we set $$z = x'$$ and get the two equations:

$z' = f(x,z,t) ~ \text{and} ~ x' = z$

Starting with initial values for $$y$$ and $$y'$$ we can produce a left hand rule approximate solutions to these equations by keeping track of $$y, z$$ and $$z'$$ as $$t$$ increases by some small increment $$d$$. We can plot solutions in three ways, as "orbits" using $$x$$ and $$z$$ as axes, or plot $$x$$ and/or $$z$$ as functions of $$t$$.

The example of forced harmonic motion:

$mx'' = -fx' - kx + vsin(\omega t)$

gives rise to the coupled equations;

$mz' = -fz -kx + vsin(\omega t) ~ \text{and} ~ x' = z$

Newton's Laws of motion yield second order differential equations for the positions of objects. There are three dimensions of motion for each particle. They are often reformulated as twice as many first order differential equations, in almost the same way. We will describe this reformulation in one dimension The same thing can be done with any number of dimensions.

In many interesting situations energy is conserved. Energy does not appear in Newton's equation $$F = ma$$. We first have to define it.

The kinetic energy of an object of mass $$m$$ moving in one dimension with speed $$v$$ is $$\frac{mv^2}{2}$$. Its momentum, $$p$$, is $$mv$$. $$p$$ rather than $$v$$ is the second variable introduced to reduced the equation to first order.

The kinetic energy is then $$\frac{p^2}{2m}$$. The force $$F$$ on the particle is defined to be the negative of the derivative of the potential energy with respect to the dependent variable (keeping all the other dependent variables and momenta fixed). Thus in the case of gravity on the surface of the earth, the force on an object of weight $$m$$ exerted by the earth is $$-mg$$, and the potential energy is $$mgh$$.

The energy also called the Hamiltonian of the system and written as $$H$$, is the sum of the kinetic and potential energies. (Incidentally, the $$H$$ symbol originally was a Greek capital eta and was chosen to be so because energy begins with E.)

Thus for gravity on the earthâ€™s surface the Hamiltonian is given by.

$H = \frac{p^2}{2m} + mgh$

The equations of motion equivalent to $$F=ma$$ then become:

$\frac{dh}{dt} = \frac{\partial H}{\partial p} ~ \text{, and} ~ \frac{dp}{dt} = -\frac{\partial H}{\partial h}$

The quaint symbols $$\frac{\partial H}{\partial p}$$ that appear here mean that you take the derivative of $$H$$ with respect to $$p$$ treating the other dependent variable $$h$$ as a constant. This sort of derivative is called the partial derivative of $$H$$ with respect to $$p$$. (In complicated situations, when there are several possible other dependent variables, its meaning depends on which ones you are keeping constant. Here it is well defined.)

Exercise 19.4 What is the Hamiltonian for an undamped and unforced harmonic oscillator (for which the force is $$-kx$$?