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Symplectic integration of Hamiltonian systems

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Abstract

We survey past work and present new algorithms to numerically integrate the trajectories of Hamiltonian dynamical systems.These algorithms exactly preserve the symplectic 2-form, i.e. they preserve all the Poincar6 invariants. The algorithms have been tested on a variety of examples and results are presented for the Fermi-Pasta-Ulam nonlinear string, the Henon-Heiles system, a four-vortex problem, and the geodesic flow on a manifold of constant negative curvature. In all cases the algorithms possess long-time stability and preserve global geometrical structures in phase space.

FAQs

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What advantages do symplectic integration algorithms (SIAs) provide over standard methods?add

The study reveals that SIAs preserve phase-space structures and Poincaré invariants over long-term simulations, unlike standard methods which often fail in these aspects. Additionally, SIAs exhibit stability properties superior to traditional integrators in various Hamiltonian systems, such as the Fermi-Pasta-Ulam problem.

How do symplectic integrators perform in chaotic Hamiltonians compared to linear systems?add

The research indicates that SIAs accurately replicate the topological structure in fully chaotic Hamiltonians, as seen in the geodesic flow of a two-holed doughnut. In contrast, while linear systems maintain numerical stability, they do not exhibit the same breadth of structural preservation as chaotic systems.

What is the significance of the Hamilton-Jacobi equation in generating SIAs?add

The Hamilton-Jacobi equation is pivotal as it allows derivation of symplectic maps through generating functions, facilitating the construction of SIAs to a desired order. By matching numerical solutions with those produced from generating functions, researchers have achieved increased accuracy and stability in simulations.

How does the Fermi-Pasta-Ulam problem illustrate the effectiveness of SIAs?add

The Fermi-Pasta-Ulam problem demonstrates that SIAs produce second-order error convergence, which was validated through comparisons with standard fourth-order Runge-Kutta methods. Despite initial lower accuracy, the SIAs showed bounded energy error growth, asserting long-term stability and global structure preservation.

What methodologies confirm the order of symplectic integrators during simulations?add

Order confirmations were achieved through numerical tests, as seen in the Fermi-Pasta-Ulam example where error scaling indicated a consistent second-order performance. Similar methodologies were employed across multiple examples, reinforcing the reliability and structural integrity of SIAs in various dynamics.

Figures (16)
Figure 1. Comparison of relative energy error of a second-order sIA and a fourth-order RKI for the Fermi-Pasta—Ulam system of oscillators. The timestep for the siA was 0,001 and 0.02 for the RKI and the final time was 7500. ken SOE The resulting algorithm is second order in the timestep. Some call the order of an algorithm the order of the error after a single timestep. We, however, will call the order of an algorithm the order of the error to reach a fixed time; this is one order less than the single timestep order. We numerically tested the order of the algorithm with the results shown in table 1. As can be seen, the algorithm is confirmed to be of second order.
Figure 1. Comparison of relative energy error of a second-order sIA and a fourth-order RKI for the Fermi-Pasta—Ulam system of oscillators. The timestep for the siA was 0,001 and 0.02 for the RKI and the final time was 7500. ken SOE The resulting algorithm is second order in the timestep. Some call the order of an algorithm the order of the error after a single timestep. We, however, will call the order of an algorithm the order of the error to reach a fixed time; this is one order less than the single timestep order. We numerically tested the order of the algorithm with the results shown in table 1. As can be seen, the algorithm is confirmed to be of second order.
rotated 90° about the q, axis, of this same object reveals the discrete D? symmetry, ie. invariance under 120° rotations and reflections due to the D° invariance of the potential. Notice also the precisely closed precessional behaviour in this perspective and the fact that the SIA achieves this by maintaining the D> symmetry of the exact system. In figure 6(d), a close-up from a different perspective of the central loop in figure 6(b) was made from runs using timesteps of dt = 0.3 (yellow) and dt = 0.15 (blue). This figure shows that halving the time step produces a structure that is éssentially the same, differing only in certain local details. In addition, closer inspection using the interactive capability of the PS-300 reveals that the trajectory triply intertwines with itself forming an intricate local structure that would not result were it not for the global properties of the SIA.
rotated 90° about the q, axis, of this same object reveals the discrete D? symmetry, ie. invariance under 120° rotations and reflections due to the D° invariance of the potential. Notice also the precisely closed precessional behaviour in this perspective and the fact that the SIA achieves this by maintaining the D> symmetry of the exact system. In figure 6(d), a close-up from a different perspective of the central loop in figure 6(b) was made from runs using timesteps of dt = 0.3 (yellow) and dt = 0.15 (blue). This figure shows that halving the time step produces a structure that is éssentially the same, differing only in certain local details. In addition, closer inspection using the interactive capability of the PS-300 reveals that the trajectory triply intertwines with itself forming an intricate local structure that would not result were it not for the global properties of the SIA.
and Neri reveals that it is just a singular projection of a standard 2-torus. Of course, not all structures are remnants of integrable objects, so not alt will fail into Fomenko’s classification; figure 6 might be one example of a non-integrable object. Nevertheless, many objects can be identified as belonging to one of Fomenko’s categories. acaisy IU WL Nil ANNES RAEN OD RE thee EN ee A EIN OO UE tw, Paralleling the sequence of figures 4(a,b), we demonstrate in figure 8 the ability of the SIA to distinguish the different global objects that can result from slightly different initial conditions. For 5000 timesteps of 6t = 0.3, we integrated twenty trajectories with initial values evenly spaced from (p,,q,) = (0,0) to (ps, q,) = (0,-0.1). We have chosen six representative trajectories, beginning in figure 8(a), with simple transitive flow on a 2-torus. In figure 8(b), it appears that the flow on the torus has coalesced onto a periodic orbit; but in fact, by linear interpolation of successive points, we see that
and Neri reveals that it is just a singular projection of a standard 2-torus. Of course, not all structures are remnants of integrable objects, so not alt will fail into Fomenko’s classification; figure 6 might be one example of a non-integrable object. Nevertheless, many objects can be identified as belonging to one of Fomenko’s categories. acaisy IU WL Nil ANNES RAEN OD RE thee EN ee A EIN OO UE tw, Paralleling the sequence of figures 4(a,b), we demonstrate in figure 8 the ability of the SIA to distinguish the different global objects that can result from slightly different initial conditions. For 5000 timesteps of 6t = 0.3, we integrated twenty trajectories with initial values evenly spaced from (p,,q,) = (0,0) to (ps, q,) = (0,-0.1). We have chosen six representative trajectories, beginning in figure 8(a), with simple transitive flow on a 2-torus. In figure 8(b), it appears that the flow on the torus has coalesced onto a periodic orbit; but in fact, by linear interpolation of successive points, we see that
Figure 6. Four views of the results from a single initial condition, (q1, p2,4q2) = (0,0, —0.1), with energy 0.117 835: (a) two-dimensional slice at the q; = 0 plane; (5) projection of the three-dimensional phase space onto the q,; = 0 plane; (c) projection of the three- dimensional phase space onto the p2 = 0 plane; (d) Comparison of results using timesteps of dt = 0.3 (yellow) and 6t = 0.15 composed twice (blue).
Figure 6. Four views of the results from a single initial condition, (q1, p2,4q2) = (0,0, —0.1), with energy 0.117 835: (a) two-dimensional slice at the q; = 0 plane; (5) projection of the three-dimensional phase space onto the q,; = 0 plane; (c) projection of the three- dimensional phase space onto the p2 = 0 plane; (d) Comparison of results using timesteps of dt = 0.3 (yellow) and 6t = 0.15 composed twice (blue).
Figure 7. The 5000 timesteps using a fifth-order sia at two different initial conditions for the Hénon—Heiles system with timestep dt = 0.3. One initial condition was (q), p2.q2) = (0,0.1,0.15) with an energy of 0.1, while the other was at (q1,p2,q2) = (0,0,0.2) with an energy of 0.117 835.
Figure 7. The 5000 timesteps using a fifth-order sia at two different initial conditions for the Hénon—Heiles system with timestep dt = 0.3. One initial condition was (q), p2.q2) = (0,0.1,0.15) with an energy of 0.1, while the other was at (q1,p2,q2) = (0,0,0.2) with an energy of 0.117 835.
separate strand. In figure 8(e), the structure has almost collapsed onto a periodic orbit, but in figure 8(f), it has re-emerged as a broad structure similar to figure 8(d).
separate strand. In figure 8(e), the structure has almost collapsed onto a periodic orbit, but in figure 8(f), it has re-emerged as a broad structure similar to figure 8(d).
Table 2. Errors for the exact system constants in the four-vortex problem using a second- order SIA. Similar behaviour was found using the algorithms of section 2 for a two-degree-of- freedom quartic oscillator with parameters chosen to make the total angular momentum a constant; the angular momentum, a non-Cartesian function, was preserved to within round-off. Application to the three-body problem preserves the nine integrals corre- sponding to centre-of-mass, centre-of-velocity, and three angular momenta to within round-off. Marsden [22] has shown that this is a general feature and that the approx- imate map on the full phase space is a lift of an approximate map on the reduced space, T°Q/G, where T°Q is the cotangent bundle of configuration space and G is the Lie group of symmetries of the Hamiltonian.
Table 2. Errors for the exact system constants in the four-vortex problem using a second- order SIA. Similar behaviour was found using the algorithms of section 2 for a two-degree-of- freedom quartic oscillator with parameters chosen to make the total angular momentum a constant; the angular momentum, a non-Cartesian function, was preserved to within round-off. Application to the three-body problem preserves the nine integrals corre- sponding to centre-of-mass, centre-of-velocity, and three angular momenta to within round-off. Marsden [22] has shown that this is a general feature and that the approx- imate map on the full phase space is a lift of an approximate map on the reduced space, T°Q/G, where T°Q is the cotangent bundle of configuration space and G is the Lie group of symmetries of the Hamiltonian.
Figure 9. Plot of (¢1,q2) scaled radially at intervals of At = 1 for the geodesic flow on a two-holed doughnut from a fourth-order sia. The timestep was dt = 0.001 and the energy was 0.2. The 5000 points plotted have the uniform distribution one expects for this system.
Figure 9. Plot of (¢1,q2) scaled radially at intervals of At = 1 for the geodesic flow on a two-holed doughnut from a fourth-order sia. The timestep was dt = 0.001 and the energy was 0.2. The 5000 points plotted have the uniform distribution one expects for this system.
of any integrator decreases exponentially with time, meaning that a certain number of bits of accuracy are lost per timestep. After a relatively short time, any integrator will have errors in the positions that are as large as the entire manifold. Thus, if one knows a priori that one’s system is Anosov there is no structural reason to choose one integrator over another, and one can simply use the most convenient integrator. However, the number of cases that are rigorously known to be completely chaotic is small and if any remnant of non-chaotic behaviour is suspected, we will argue in section 5 that there are good reasons to use a SIA. Let us note in passing an interesting aspect of this system. All Hamiltonian systems are locally integrable [28]. For this system the local integrals are obvious, namely, as long as the trajectory remains inside the fundamental polygon, both the energy and angular momentum are constant. When the trajectory crosses the boundary of the fundamental polygon, the energy is unchanged by the transformation (22)-(24), but
of any integrator decreases exponentially with time, meaning that a certain number of bits of accuracy are lost per timestep. After a relatively short time, any integrator will have errors in the positions that are as large as the entire manifold. Thus, if one knows a priori that one’s system is Anosov there is no structural reason to choose one integrator over another, and one can simply use the most convenient integrator. However, the number of cases that are rigorously known to be completely chaotic is small and if any remnant of non-chaotic behaviour is suspected, we will argue in section 5 that there are good reasons to use a SIA. Let us note in passing an interesting aspect of this system. All Hamiltonian systems are locally integrable [28]. For this system the local integrals are obvious, namely, as long as the trajectory remains inside the fundamental polygon, both the energy and angular momentum are constant. When the trajectory crosses the boundary of the fundamental polygon, the energy is unchanged by the transformation (22)-(24), but

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