A Lie group formalism for global parameter space mapping of ordinary differential equations, described in the preceding paper, is developed further here. The need for such mapping arises in a variety of physical contexts. The procedure is demonstrated on, but not restricted to, the system of coupled equations Ẋ = cX. The Lie group generators are obtained exactly from both the time-independent and time-dependent generating equations. The transformations obtained from these generators leave the system of differential equations invariant. The time-independent transformations map any solution of the linear system Ẋ = cX into any other solution with the same frequency or time constant. The time-dependent transformations interconvert solutions with different frequencies and/or time constants. Any solution of Ẋ = cX can be mapped into any solution of Ẋ = c′X. Thus the behavior of X (t) can be examined as a function of changes in any of the system parameters, or in the initial conditions, X(0). As an example, one of the time-dependent mappings demonstrates the continuous transformation of oscillator solutions into nonoscillatory ones, and vice versa.
All Science Journal Classification (ASJC) codes
- Physical and Theoretical Chemistry