The spring group

Does the ``spring group'' form a group in the formal mathematical sense? A group is a set of elements and a binary operator ``+'' that satisfies four conditions (Herstein, 1975):

• [1)] Closure: A and B in the group implies A+B in the group.
• [2)] Associativity: A, B, C in the group implies that A + (B+C) = (A+B) +C.
• [3)] Identity: There exists an element E in the group such that A + E = E + A = A for all A in the group.
• [4)] Inverse: For every element A in the group there exists another element $A_{\mbox{\rm\scriptsize inverse}}$ in the group such that $A + A_{\mbox{\rm\scriptsize inverse}} = A_{\mbox{\rm\scriptsize inverse}} + A = E$.

If in addition a group also satisfies

• [5)] Commutativity: For all A and B in the group A+B = B+A.

it is said to be an Abelian Group.

For our spring example the elements in the group are springs parameterized by their compliance 1/k. The binary operator ``+'' represents connecting two springs in series and replacing the result with an equivalent single spring. Note that in terms of compliance (the ``spring-group'' representation) ``+'' behaves exactly like standard addition. Given this definition, closure, associativity, and commutativity are obvious enough. The identity element is just the infinitely stiff spring with 1/k = 0. In order to have inverses as required we must allow 1/k < 0; to get the inverse of a given spring just change the sign on the spring compliance 1/k.

It appears that if we allow springs with negative k into our set the ``spring group'' is indeed a formal Abelian group. Is there any practical reason to do so? The group notation does not seem particularly enlightening for the trivial case of springs in series, but the group concept of inverse elements is quite a useful one for more complex equivalent-medium systems such as the Schoenberg-Muir calculus. Inverse elements form the theoretical basis for layer stripping and more generally allow us to decompose a bulk equivalent (what we can actually observe) back into candidate heterogeneous models that may represent the real earth.