1. Mathematical Methods
1.1 The Hilbert (axiomatic) method
We place a locked cage onto a given point in the desert. After that we introduce the following logical system: Axiom 1: The set of lions in the Sahara is not empty. Axiom 2: If there exists a lion in the Sahara, then there exists a lion in the cage. Procedure: If P is a theorem, and if the following is holds: P implies Q', then Q is a theorem. Theorem 1: There exists a lion in the cage.
1.2 The geometrical inversion method
We place a spherical cage in the desert, enter it and lock it from inside. We then performe an inversion with respect to the cage. Then the lion is inside the cage, and we are outside.
1.3 The projective geometry method
Without loss of generality we can view the desert as a plane surface. We project the surface onto a line and afterwards the line onto an interiour point of the cage. Thereby the lion is mapped onto that same point.
1.4 The Bolzano-Weierstrass method
Divide the desert by a line running from north to south. The lion is then either in the eastern or in the western part. Lets assume it is in the eastern part. Divide this part by a line running from east to west. The lion is either in the northern or in the southern part. Lets assume it is in the northern part. We can continue this process arbitrarily and thereby constructing with each step an increasingly narrow fence around the selected area. The diameter of the chosen partitions converges to zero so that the lion is caged into a fence of arbitrarily small diameter.
1.5 The set theoretical method
We observe that the desert is a separable space. It therefore contains an enumerable dense set of points which constitutes a sequence with the lion as its limit. We silently approach the lion in this sequence, carrying the proper equipment with us.
1.6 The Peano method
In the usual way construct a curve containing every point in the desert. It has been proven  that such a curve can be traversed in arbitrarily short time. Now we traverse the curve, carrying a spear, in a time less than what it takes the lion to move a distance equal to its own length.
1.7 A topological method
We observe that the lion possesses the topological gender of a torus. We embed the desert in a four dimensional space. Then it is possible to apply a deformation  of such a kind that the lion when returning to the three dimensional space is all tied up in itself. It is then completely helpless.
1.8 The Cauchy method
We examine a lion-valued function f(z). Be z the cage. Consider the integral where c represents the boundary of the desert. Its value is f(z), i.e. there is a lion in the cage .
1.9 The Wiener-Tauber method
We obtain a tame lion, , from the class , whose fourier transform vanishes nowhere. We put this lion somewhere in the desert. then converges toward our cage. According to the general Wiener-Tauner theorem  every other lion L will converge toward the same cage. (Alternatively we can approximate L arbitrarily close by translating through the desert .)
2 Theoretical Physics Methods
2.1 The Dirac method
We assert that wild lions can ipso facto not be observed in the Sahara desert. Therefore, if there are any lions at all in the desert, they are tame. We leave catching a tame lion as an execise to the reader.
2.2 The Schroedinger method
At every instant there is a non-zero probability of the lion being in the cage. Sit and wait.
2.3 The nuclear physics method
Insert a tame lion into the cage and apply a Majorana exchange operator  on it and a wild lion. As a variant let us assume that we would like to catch (for argumentıs sake) a male lion. We insert a tame female lion into the cage and apply the Heisenberg exchange operator , exchanging spins.
2.4 A relativistic method
All over the desert we distribute lion bait containing large amounts of the companion star of Sirius. After enough of the bait has been eaten we send a beam of light through the desert. This will curl around the lion so it gets all confused and can be approached without danger.
3 Experimental Physics Methods
3.1 The thermodynamics method
We construct a semi-permeable membrane which lets everything but lions pass through. This we drag across the desert.
3.2 The atomic fission method
We irradiate the desert with slow neutrons. The lion becomes radioactive and starts to diintegrate. Once the disintegration process is progressed far enough the lion will be unable to resist.
3.3 The magneto-optical method
We plant a large, lense shaped field with cat mint (nepeta cataria) such that its axis is parallel to the direction of the horizontal component of the earthıs magnetic field. We put the cage in one of the fieldıs foci. Throughout the desert we distribute large amounts of magnetized spinach (spinacia oleracea) which has, as everybody knows, a high iron content. The spinach is eaten by vegetarian desert inhabitants which in turn are eaten by the lions. Afterwards the lions are oriented parallel to the earthıs magnetic field and the resulting lion beam is focussed on the cage by the cat mint lense.