Existence and Uniqueness in Mechanics
- (The Lorentz transformation)
[Rob, p.6, #2.2] starts with the most
general transformation [Rob, p.6, l.- 2] and then
gradually eliminates the impossibilities. Therefore, it takes quite an argument
even to derive the simple fact that yN=y [Rob, p.7,
l.17-l.20]. However, Robinson’s approach is indispensable for proving the
uniqueness [Rob, p.8, l.- 1] of the Lorentz
transformation. In contrast, Eisberg proves yN=y by
a simple geometric
consideration [Eis, p.A-2, l.14]. In addition, Eisberg, uses time dilation and
length contraction to explain why the Lorentz transformation must be of a certain
form [1]. Thus Eisberg’s approach facilitates the effective construction of at least
one useful Lorentz transformation.
- Construction vs. Resources
The set of logical resources is a subset of the set of mathematical resources. The set of mathematical resources is a subset of the set of physical resources. The existence
of a solution can be more effectively constructed if we are allowed to use a bigger set of resources. For example, we show the existence of an electric field by using a charge to test its response to the field, and then measure the size of the field.
[Kara, p.166, l.−5-l.−1]
is an awkward way to say the following statement:
The construction of a dynamic solution is more effective than that of a kinematic solution, because there are more resources in dynamics than in kinematics.
- If we simply substitute [Jack, p.480, (10.84)] into [Jack, p.480, (10.80)] or [Jack, p.480, (10.82)], we will find that [Jack, p.480, (10.84)] are the required Green functions, but we still don’t know how they are constructed. The method of images [Jack, p.480, l.-3] provides an effective way to construct these Green functions.
- Jackson divides the solution of [Jack, p.180, (5.26)] into two cases: (a). J=0
[Jack, p.180, l.-16-l.-8] and (b). J¹0
[Jack, p.180, l.-7-p.181, l.22]. Case (a) is a
special case of [Kara, pp.90-91, §5.7].
Case (b) is a special case of [Kara, p.92, §5.8.2].
Thus, Karamcheti provides a broader perspective. In contrast, Jackson's
solution in case (b) is more effective than Karamcheti's because Jackson
constructs his solution using physical considerations.
Remark 1. At best, B=-ÑF_{M} in [Jack, p.180, l.-14] is only a reasonable
guess [Kara, p.85, l.-15].
Its validity can be proved by the existence of a solution of Laplace's equation
[Kara, p.91, (5.37)].
Remark 2. The first term on the right-hand side of [Jack, p.180, (5.28)] is a particular solution of [Jack, p.180, (5.27)].
All the solutions of [Jack, p.180, (5.27)] can be expressed in the form of
[Jack, p.180, (5.28)].
- Sometimes, the standardized problem which has a general solution does not
quite fit the given conditions of a practical problem
[Wangs, p.71, l.-6-l.-1]. In this case, it would very difficult to solve the problem using
the general solution. The uniqueness of the solution guarantees that if we can use
available resources to effectively find a solution, then this solution will be the
solution [Wangs, p.172, l.-6-l.-1].
Thus, the uniqueness gives us more room to improve the effectiveness of our
solving method.
Example. If we want solve the Poisson equation [Wangs. p.171, (11-2)]
physically, we must treat the divergence and curl of a field as the sources [Wangs,
p.37, l.-2-l.-1]. Using
the charge distribution within a given region and the boundary conditions, we can construct
fictitious charges outside the region so that the given boundary conditions will
be satisfied. Theoretically, we can then solve our problem by [Wangs, p.69,
(5-7)]. However, using these fictitious charges, we can find a more effective
method [Wangs, pp.173-183, §11-2] to solve
our problem.
- The uniqueness of the solution with Neumann boundary conditions implies that all the solutions that satisfy [Jack, p.155, (4.41) & (4.42)] are the same.
It does not matter whether or not the distribution of the image charges is reasonable. It does not matter whether
or not the formula for potential in terms of image
charges is valid physically. The important thing is that the final answer
(potential) must satisfy [Jack, p.155, (4.41) & (4.42)]. It is unnecessary to
trace the origin of ideas that inspire the construction. It does not matter
whether or not the construction has a physical meaning and whether or not the formula used in
the construction obeys or contradicts physics laws. Thus, there is no
contradiction even though [Wangs. p.181, (11.41)] and [Jack, p.155, (4.44)] use
different formulae to approach the same problem.