Stationary Solutions An unbelievably basic question, but it's something I've never been taught. Am I right in thinking that the following defines a stationary solution?
Let $\phi$ be some dynamical variable satisfying a differential equation $D(\phi)=0$ with respect to some parameter $t$. Then $\phi$ is called a stationary solution or a stationary state if all $t$ derivatives of $\phi$ are zero.
Could someone point me to a resource where this is defined? A quick google threw up nothing useful, but perhaps I was looking in the wrong place!
 A: Here's a quote from section 10.2 of "A first course in general relativity" by Schutz:

We define a static spacetime to be one in which we can find a time
  coordinate $t$ with two properties:  (i) all metric components are
  independent of $t$, and (ii) the geometry is unchanged by time
  reversal, $t \rightarrow -t$.
...
(A spacetime with property (i) but not necessarily (ii) is said to be
  stationary.)

So, your statement

Then ϕ is called a stationary solution or a stationary state if all t
  derivatives of ϕ are zero.

certainly seems consistent with Schutz's description of stationary.
A: I think Wikipedia has a fairly good definition for this:

A stationary state is called stationary because a particle remains in the same state as time elapses, in every observable way.

What this means is that every observable quantity which can be computed from the state is constant in time.
In classical mechanics, this is kind of trivial because the state is just given by the positions and velocities (or momenta) of all particles in the system. In other words, the state is itself an observable quantity, and therefore, as you said, a stationary state is necessarily constant. But it's not particularly common to use the term "stationary state" in classical mechanics.
Where it really becomes useful is in quantum mechanics. Here, the state is more than just position and momentum. A quantum state contains some non-observable information, and that information can change over time even if all the observable quantities are constant. So in quantum mechanics, "stationary state" does not mean that all $t$ derivatives of the state are zero. The correct mathematical criterion is that these states are eigenstates of the Hamiltonian, $H\lvert\psi\rangle \propto \lvert\psi\rangle$.
A: Independent of the interpretation of a differential equation, we have the following:
A stationary solution of an autonomous differential equation $F(y(t),\dot y(t))=0$ (not depending explicitly on time) is a solution that doesn't depend on time. 
Thus the stationary solutions are precisely the solutions of the form $y(t)=y_0$, where $y_0$ solves the nonlinear equation $F(y_0,0)=0$. One doens't need to look at higher derivatives than those in the differential equation, although all derivatives are of course zero for a constant solution).
For a second order differential equation $F(y(t),\dot y(t),\ddot y(t))=0$ the corresponding condition on $y_0$ is $F(y_0,0,0)=0$.
