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### QT's generalization to arbitrary Hamiltonian systems

If the configuration space is D dimensional, then there are 2D dimensions in the phase space. It can be shown that in a Hamiltonian system, the number B of stable and unstable directions is equal, although B < D is possible. At time , let represent D unstable unit vectors, and let represent D stable unit vectors. For any particular timestep, it will be convenient if the unstable vectors are orthonormal to each other, and the stable vectors are orthonormal to each other. However, the stable and unstable vectors together will not in general form an orthogonal system.

The vectors are evolved as in the two dimensional case, except using Gram-Schmidt orthonormalization to produce two sets of D-orthonormal vectors at each timestep. Arbitrary orthonormal bases are chosen, at time , and at time , and then evolved according to

At each step i, two Gram-Schmidt orthonormalizations are done: one on to produce , and another on to produce . After a few e-folding times, points in the most unstable direction at step i, points in the second most unstable direction, etc.  Likewise, points in the most stable direction at time i, points in the second most stable direction, etc.

The multidimensional generalization of (10) is the obvious

To convert the 1-step error at step i from phase-space co-ordinates to the stable and unstable basis , one constructs the matrix whose columns are the unstable and stable unit vectors, , and solves the system .

From (8) and (16) the equation for the correction co-efficients in the unstable subspace at step i+1 is

which are projected out along producing

where the scalar , and the Gram-Schmidt process ensures if j < k. As stated previously, the boundary condition requires the unstable component of the corrections to be small at timestep S, and their stable components to be small at timestep 0. For simplicity we take , as did QT, and the co-efficients are computed backward using

We first solve for , which does not require knowledge of the other co-efficients, then solve for , etc.

Again from (8) and (16), the equation for the correction co-efficients in the stable subspace at timestep i is

which are projected out along producing

where , and the Gram-Schmidt process ensures if j < k. The boundary condition at step 0 is , and the co-efficients are computed forward using

As with the unstable corrections, we first compute which does not require knowledge of the other co-efficients, then we compute , etc.

Next: Discussion and Results of Up: Refinement of a numerical Previous: Refinement of a numerical

Wayne Hayes
Fri Dec 27 17:41:39 EST 1996

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