Answer to the Question 10/02
LEAST ACTIONThe question was:
In analytical mechanics Newton's laws are reformulated as Hamilton's Principle of Least Action. The actual derivation of Newton's laws (via the Euler-Lagrange equations) requires only that the classical path be a stationary point of the action functional. But the term "Least Action" seems to imply that the path must be a minimum of the functional.
Can you think of examples where the classical path is a local maximum of the action? If not, how about a saddle point?
(1/2003) On 29/12/2002 Gleb Gribakin from the Department of Applied Mathematics and Theoretical Physics Queen's University Belfast, U.K. (email g.gribakin@qub.ac.uk) suggested the following example:
Suppose the particle is confined to the surface of a sphere and the potential energy on this surface is constant, i.e. the particle moves freely on the surface. The path of the particle then lies on a circle whose diameter is equal to that of the sphere. For any two points on such a circle that are not diametrically opposite, say 1 and 2, there are two possible paths which start at 1 and end at 2. The shorter arc is a local (and global) minimum of the action integral. The longer one must be a saddle point. A variation of the path, which keeps the midpoint of the longer arc fixed, increases the action. On the other hand, if we allow the midpoint to move in the direction perpendicular to the original long arc, the action integral can be made smaller (together with the length of the path).
Benjamin Svetitsky who originally suggested this problem offered an additional example of saddle point, which can be found in the following file.
David Augier (email daugier@yahoo.fr suggested (28/1/03) the following example in which and extremum (in Fermat's principle of least time in optics) does not correspond to minimum:
Consider an ellipsoidal (concave) mirror. As everybody know, its two foci, A and B, verify AM+BM=constant for every M on the surface of the ellipsoid. As a consequence, every AMB is a light ray.
Let M be a certain point on the surface of the ellipsoid. Now, forget this ellipsoidal mirror (just keep its shape in mind) and consider a concave mirror, which is tangent to the ellipsoid at M, and strictly included inside the ellipsoid (except M of course). As this new mirror is tangent to the ellipsoid, AMB is still a light ray (Snell-Descartes laws are still valid). However, every ANB (N belongs to the surface of the concave miror), with N quite near to M, is clearly smaller than AMB, and is not a light ray (still because of Snell-Descartes laws). AMB is a local maximum.
If the concave mirror is stricly included outside the ellipsoid, the light ray is a local minimum.
And if you consider a mirror with a shape like x3 (still tangent to the ellipsoid), a part of the mirror is inside the ellipsoid, and another part is outside. Then AMB is only stationnary (saddle point).
Y. Kantor: We are waiting for more examples and discussions.
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