Differential algebraic methods for obtaining approximate numerical solutions to the Hamilton-Jacobi equation
I present two differential-algebraic (DA) methods for approximately solving the Hamilton- Jacobi (HJ) equation. I use the “automatic differentiation” property of DA to convert the nonlinear partial-differential HJ equation into a initial-value problem for a DA-valued first-order ordinary differential equation (ODE), the “HJ/DA equation”. The solution of either form of the HJ/DA equation is equivalent to a perturbative expansion of Hamilton’s principle function about some reference trajectory (RT) through the system. The HJ/DA method also extracts the equations of motion for the RT itself. Hamilton’s principle function generates the canonical transformation, or mapping, between the initial and final state of every trajectory through the system. Since the map is represented by a generating function, it must automatically be symplectic, even in the presence of round-off error.
The DA-valued ODE produced by either form of HJ/DA is equivalent tc a hierarchically-ordered system of real-valued ODEs without “feedback” terms; therefore the hierarchy may be truncated at any (arbitrarily high) order without loss of self consistency. The HJ/DA equation may be numerically integrated using standard algorithms, if all mathematical operations are done in DA. I show that the norm of the DA-valued part of the solution is bounded by linear growth. The generating function may be used to track either particles or the moments of a particle distribution through the system.
In the first method, all information about the perturbative dynamics is contained in the DA-valued generating function. I numerically integrate the HJ/DA equation, with the identity as the initial generating function. A difficulty with this approach is that not all canonical transformations can be represented by the class of generating functions connected to the identity; one finds that with the required initial conditions, the generating function becomes singular near caustics or foci. One may continue integrating through a caustic by using a Legendre transformation to obtain a new (but equivalent) generating function which is singular near the identity, but nonsingular near the caustic. However the Legendre transformation is a numerically costly procedure, so one would not want to do this often. This approach is therefore not practical for systems producing periodic motions, because one must perform a Legendre transformation four times per period.
The second method avoids the caustic problem by representing only the nonlinear part of the dynamics by a generating function. The linearized dynamics is treated separately via matrix techniques. Since the nonlinear part of the dynamics may always be represented by a near-identity transformation, no problem occurs when passing through caustics.
I successfully verify the HJ/DA method by applying it to three problems which can be solved in closed form. Finally, I demonstrate the method’s utility by using it to optimize the length of a lithium lens for minimum beam divergence via the moment-tracking technique.