Linear Representation

1. For fixed $\mathbf{I}=(I_u,I_v,I_x,I_y)$ these paths have the linear representation

$$\phi^u =\phi^u_0 +(\tau-\tau_0)\frac{\partial H(\mathbf{I})}{\partial I_u}$$ (47)

$$\phi^v =\phi^v_0 +(\tau-\tau_0)\frac{\partial H(\mathbf{I})}{\partial I_v}$$ (48)

$$\phi^x =\phi^x_0 +(\tau-\tau_0)\frac{\partial H(\mathbf{I})}{\partial I_x}$$ (49)

$$\phi^y =\phi^y_0 +(\tau-\tau_0)\frac{\partial H(\mathbf{I})}{\partial I_y}$$ (50)

It is related to the spacetime representation relative to the physically given coordinates $x^\alpha=(u,v,x,y)$ by the transformation

$$\left.\begin{array}{rl} \phi^u&=\displaystyle \frac{\partial W(u,... ...style \frac{\partial W(u,v,x,y;\mathbf{I})}{\partial I_y}~. \end{array}\right\}$$ (51)

This transformation has changed a nonlinear representation into an equivalent representation which is linear.