6.3 The Universal Graph Method

We may regard the Theorem $\pi$ as the specific construction in dimensional geometry. It transfers any dimensional dependence onto some function at the dimensionless fiber. This is the most important gain from the Theorem $\pi$. However there is also a price to pay. The construction is not unique and some ambiguity appears due to the dimensional basis.

Let us now present the competitive method. It also reduces the dimensional dependencies onto some functions at the dimensionless fiber but it does not break the symmetry among incoming arguments. We have called it the universal graph method. At first we shall illustrate our idea by an example.

Example 6.2   Let us recollect the problem presented in the example 6.1.

Figure: a) The construction in the dimensional space. b) The form of universal graph $\cal {G}$ in the space of models X.

At first we note that both constructions in Fig. 6.1 make use of the specific point 1 at the dimensionless fiber. Moreover, the points b, c form Fig. 6.1 a) were treated in the same way in the fixed base a. However the point c was supposed to be the value, whereas a, b are arguments of the dimensional function.

In Fig. 6.2 a) we have depicted another geometrical construction. We have drawn two lines ac and bc which cross the dimensionless fiber at the points $\xi_{a}^{}$ and $\xi_{b}^{}$ respectively. The equations of lines are

ac :    a^{1 - t}c^t    t $$\in$$ R        and        bc :    b^{1 - u}c^u    u $$\in$$ R.

After simple calculations, taking into account the dimensions from (6.5) we get

$$\xi_{a}^{}$$ = c^-2a³    (t = - 2)        and        $$\xi_{b}^{}$$ = c^{- ${\textstyle\frac{1}{2}}$}b^{${\textstyle\frac{3}{2}}$}    (u = - $${\textstyle\frac{1}{2}}$$).

Therefore, the points c,$\xi_{a}^{}$ and c,$\xi_{b}^{}$ are not independent for any given a, b.

We may think that the line ac is given by the points a,$\xi_{a}^{}$ and similarly, the pair of points b,$\xi_{b}^{}$ defines the line bc. Next we impose the consistency condition. The lines a,$\xi_{a}^{}$ and b,$\xi_{b}^{}$ should cross the value fiber at one common point c. The direct calculations yield

$$\xi_{a}^{-\frac{1}{2}} {\textstyle\frac{3}{2}} \xi_{b}^{-2}$$ (6.6)

Now the dimensionless points $\xi_{a}^{}$ and $\xi_{b}^{}$ become connected within the orbit equation

$$\xi_{a}^{-\frac{1}{2}}$$a^{${\textstyle\frac{3}{2}}$} = $$\xi_{b}^{-2}$$b³.

For a^{${\frac{3}{2}}$}/b³ = const $\in$ $\bf R_{+}^{}$ the ratio $\xi_{b}^{-2}$/$\xi_{a}^{-\frac{1}{2}}$ is also constant and fixed. The real positive number a^{${\frac{3}{2}}$}/b³ labels the orbit of arguments with respect to the action of the group $\it Hom$($\cal {W}$,$\bf R_{+}^{}$).

We involve the space of models X = $\bf R_{+}^{}$ x $\bf R_{+}^{}$ with coordinate axes $\xi_{a}^{}$,$\xi_{b}^{}$ as is depicted in Fig. 6.2 b). Then each curve

$${\frac{\xi_{b}^{-2}}{\xi_{a}^{-\frac{1}{2}}}} \kappa \in \bf R_{+}^{}$$ (6.7)

corresponds to some orbit of arguments a, b. We obtain the geometrical representation of any orbit of arguments!

We summarize the important features of the above orbit representation.

1^o.

The curve (6.7) (as geometrical manifold) does not depend on the concrete choice of the orbit parameter labelling the orbits of arguments a, b.

2^o.

The curve (6.7) requires p(c) to be fixed, but it is independent of the concrete value of c.

We conlude that we have divided the space of models X into disjoint subsets (manifolds) representing the orbits of arguments a, b.

Supposing now that the orbit is fixed for the moment. Therefore we have the concrete curve $\kappa_{0}^{}$ in the space of models. The value c₀ corresponds to the arguments a₀, b₀ from this fixed orbit: c₀ = c(a₀, b₀). Any other point a, b from this orbit can be achieved with a suitable action of the group $\it Hom$($\cal {W}$,$\bf R_{+}^{}$). If $\eta$ ( $\eta$ $\in$ $\bf R_{+}^{}$) denotes the scaling factor of b then the scaling factors for a, c are

b = $$\eta$$b₀,        a = $$\eta^{2}_{}$$a₀,        c = $$\eta^{3}_{}$$c₀,

according to the dimensions from (6.5). We obtain

c = $$\eta^{3}_{}$$c₀ = $$\xi_{a}^{-\frac{1}{2}}$$$$\left(\vphantom{\eta^{2}a_{0}}\right.$$$$\eta^{2}_{}$$a₀$$\left.\vphantom{\eta^{2}a_{0}}\right)^{\frac{3}{2}}_{}$$ = $$\xi_{b}^{-2}$$$$\left(\vphantom{\eta b_{0}}\right.$$$$\eta$$b₀$$\left.\vphantom{\eta b_{0}}\right)^{3}_{}$$.

Therefore all factors including $\eta$ cancel and we arrive at the equation

c₀ = $$\xi_{a}^{-\frac{1}{2}}$$a^{${\textstyle\frac{3}{2}}$}₀ = $$\xi_{b}^{-2}$$b³₀.

We conclude that inside the single orbit of arguments a, b (or geometrically along the single curve (6.7) and for the given dimensional function c(a, b) the values of $\xi_{a}^{}$ and $\xi_{b}^{}$ are constant.

We have proved that the invariant function is fixed by its values at one single point inside each orbit for all orbits of arguments. The construction described above manifests this property at the geometrical level.

The collection of above single points at each curve in the space of models will be called the universal graph $\cal {G}$ of the function c(a, b). The whole construction is depicted in Fig. 6.2 b).

An essential fact must be stressed. The universal graph represents the dimensional function in a unique geometrical way, independently of any particular choice of orbit parameters.

Assuming that the number labelling different orbits (or equivalently curves (6.7) in X) is chosen as in (6.7):

$${\frac{\xi_{b}^{-2}}{\xi_{a}^{-\frac{1}{2}}}}$$ = $$\kappa$$ = $${\frac{a^{\frac{3}{2}}}{b^{3}}}$$.

Any point at $\cal {G}$ has the two coordinates $\xi_{a}^{}$ and $\xi_{b}^{}$ in the space of models X. The functions $\xi_{a}^{}$($\kappa$), $\xi_{b}^{}$($\kappa$)

c = $$\xi_{a}^{-\frac{1}{2}}$$($$\kappa$$)a^{${\textstyle\frac{3}{2}}$} = $$\xi_{b}^{-2}$$($$\kappa$$)b³

restore the dimensional dependence c(a, b) in the dimensional bases a and b respectively.

Our idea consists in the estimation and approxiomation of the universal graph for any dimensional function. Then there is no problem of the dimensional bases. Moreover we do not break the symmetry among incoming arguments.

In our example we know the form of c(a, b) = ab. From (6.6) we easily get

c = ab = c^{${\textstyle\frac{2}{3}}$}$$\xi_{a}^{\frac{1}{3}}$$c^{${\textstyle\frac{1}{3}}$}$$\xi_{b}^{\frac{2}{3}}$$,

or equivalently

$$\xi_{a}^{}$$$$\xi_{b}^{2}$$ = 1.

This also demonstrates some general property. The equation for the universal graph $\cal {G}$ does not include the orbit parameter $\kappa$. From experimental values of a, b, c we may evaluate $\xi_{a}^{}$ and $\xi_{b}^{}$. Next we seek the equation for $\cal {G}$ in the form

F($$\xi_{a}^{}$$,$$\xi_{b}^{}$$) = 1.

In practice this step requires an approximation technique. Having found the universal graph $\cal {G}$ we cross it with each orbit to restore the form of the functional dependence.

Similarly as in the point 6.2 we search for the form of the dimensional function q(q₁,..., q_n). The point 1 at the dimensionless fiber loses it's privileged position and will be omitted in our constructions.

Note that the method presented in the Example 6.2 is not completely general. A few different types of constructions may appear. They are all presented in examples introduced in the next point.

Generaly each construction yields the equation of the form

q = $$\xi_{i_{1}...i_{l}}^{\alpha_{0}}$$ x q_i1^$\alpha_{1}$ x ... x q_il^$\alpha_{l}$,

where $\alpha_{0}^{}$,$\alpha_{1}^{}$,...,$\alpha_{l}^{}$ $\in$ R and l $\leq$ k. Next we impose the consistency condition. All constructions should yield the same point at the value fiber.

Then we will obtain the set of equations of the form

$$\xi_{i_{1}...i_{l}}^{\alpha_{0}}$$ x qi1$\alpha_{1}$ x ... x qil$\alpha_{l}$ = $$\xi_{j_{1}...j_{l^{\prime}}}^{\beta_{0}}$$ x qj1$\beta_{1}$ x ... x qjl$\prime$$\beta_{l^{\prime}}$.

In general formulations the indices become somewhat complicated but in practical applications this problem disappears. This is why we prefer to present our method within examples. The general form becomes crabbed. The index should include the type of construction and the information about arguments appearing in the given construction. All that will appear as an index labelling the power exponents!

How many different constructions should be taken into account? It depends on the current problem. The general criterion says that the suitable ratios

$${\frac{q_{i_{1}}^{\alpha_{1}} \times ... \times q_{i_{l}}^{\alpha... ...{j_{1}}^{\beta_{1}} \times ... \times q_{j_{l^{\prime}}}^{\beta_{l^{\prime}}}}}$$

should uniquely label the orbits of arguments.

Next collecting all $\xi$ with different indices we form the space of models X. In general cases X = $\bf R_{+}^{N}$ = $\bf R_{+}^{}$ x ... x $\bf R_{+}^{}$ (x denotes the Cartesian product).

The structure of orbits is obtained from the consistency conditions. Each orbit is represented by a curve in X. Note however that orbits need not fill the whole space X. The curves may give only some specific manifold in X.

The concrete value q of the dimensional function q(q₁,..., q_n) corresponds to one, single point at each orbit. It means that at any curve (representing orbit) we have exactly one point. The collection of all such points is called the universal graph $\cal {G}$ of the dimensional function.

Geometrically the universal graph is a specific cross-section of the manifold of orbits. It represents the dimensional function but with no particular dimensional basis. To restore the form coming from the Theorem $\pi$ one should project $\cal {G}$ onto a chosen axis in the space of models X.

We agree that the construction of $\cal {G}$ looks strange and complicated. To ensure the reader that our method is as effective as the classical Theorem $\pi$ a few more advanced examples will be presented subsequently followed by the general algorithm.