P. 236(12) 选作题

This problem outlines a proof of the Buckingham pi theorem, which states (roughly speaking) that any complete equation implies an equation in which all variables appear in dimensionless combinations.

(a) Let  be measurements of  (primary or secondary) quantities. Suppose that there is one and only one functional relationship

                                                                                               (1)

connecting these measurements and that equation (1) is complete. Suppose further that there are  fundamental units and that each is decreased by a factor . Show that there must be a relationship of the form

.                                                    (2)

(As the proof unfolds, it may be helpful to work a particular example that illustrates the general results for which the exercise calls.)

(b) Deduce from equation (2) that

,                                                                                (3)

where a subscript  on  denotes the partial derivative with respect to the ith argument.

(c) Assuming that , introduce the new independent variables ,

,                                                                                (4)

so that

,                                                                                              (5)

where

,                                                                        (6)

Show that

,                                                                                                    (7)

where the subscript  on  denotes a derivative with respect to the kth argument. Show also that all quantities  have dimensional exponent unity with respect to the first primary quantity.

(d) Introduce another new set of independent variables :

,                                                             (8)

so that

,                                                                                             (9)

where

,                                            (10)

Show that . Show also that the quantities  are dimensionless with respect to the first primary quantity. Thus equation (1) implies equation (9), a relation involving m-1 quantities that have a dimensional exponent zero with respect to the first primary quantity.

(e) Show that the above conclusion remains valid even if some or all of the quantities  are zero.

(f) Deduce finally the relation

,                                                                                            (10)

where the quantities  are dimensionless. Noting the change of variables involved, show also that the  are products of powers of . Show that “usually” there will be m-n such dimensionless products.

REMARK. Suppose that  can be solved for  and that the expression for  actually contains . If one solves for , then each term in the resulting expression must be of the same dimension. This result is called the principle of dimensional homogeneity.

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