STEP-FUNCTIONS 



22/2 



If the elastic breaks, k becomes 0, and the equations become 



dx 1 



dt 

 dx 2 

 ~dt 



(3) 



Assume that the elastic breaks if it is pulled longer than X. 



The events may be viewed in two ways, which are equivalent. 



We may treat the change of k as a change of parameter to the 

 2-variable system x v x 2 , changing their equations from (2) 

 above to (3) (S. 21/1). The field of the 2-variable system will 

 change from A to B in Figure 22/2/1, where the dotted line at X 



A B 



Figure 22/2/1 : Two fields of the system {x x and x 2 ) of S. 22/2. 

 unbroken elastic the system behaves as A, with broken as B. 

 the strand is stretched to position X it breaks. 



With 

 When 



shows that the field to its right may not be used (for at X the 

 elastic will break). 



Equivalent to this is the view which treats them as a 3- 

 variable system : sc l9 x 2 , an d k. This system is absolute, and has 

 one field, shown in Figure 22/2/2. 



In this form, the step-function must be brought into the 

 canonical equations. A possible form is : 



dk (K K 

 dt = q [-2 +2 



where K is the initial value of the variable k, and q is large and 

 positive. As q— ■> oo, the behaviour of k tends to the step- 

 function form. 



Another method is to use Dirac's ^-function, defined by S(u) = 

 if u ?±0, while if u = 0, d(u) tends to infinity in such a way that 



rco 



I d(u)du = 1. 



J —00 



233 



+ - tanh {q(X - x,)} - k 



(4) 



