Symplectic Regularization Step

• Symplectic regularization step is a modification that preserves the canonical Hamiltonian structure in systems with variable or phase-space-dependent time-stepping.
• By embedding time into an extended phase space or using non-canonical generating functions, these methods restore error boundedness and long-term conservation of invariants.
• Numerical experiments confirm that optimized adaptive time-stepping via backward error analysis minimizes secular drift and resonance instabilities in Hamiltonian simulations.

A symplectic regularization step refers to a procedural or algorithmic modification that either restores, preserves, or adapts the underlying symplectic (canonical Hamiltonian) structure in contexts where this structure is compromised—such as during variable time-stepping in symplectic integrators, transformations that destroy canonical form, or adaptive numerical schemes for Hamiltonian systems. The primary goal is to eliminate pathological behaviors (unphysical growth, secular errors, parametric instabilities) and guarantee long-term qualitative correctness by ensuring that the geometric invariants of the original problem are maintained or appropriately modified. In the context of adaptive symplectic integration, symplectic regularization underpins the design of integrators that can robustly handle changes in time-step selection, especially when such changes depend on phase-space variables.

1. Pathologies in Variable Time-Step Symplectic Integrators

Variable time-step symplectic integrators exhibit two distinct sources of pathologies, as established via backwards error analysis:

• Time-dependent step, $\Delta = \Delta(t)$: Even when a symplectic method is used, the modified (shadow) Hamiltonian acquires explicit periodic time-dependence. For Hamiltonians such as $H = \frac{1}{2}(p^2+\Omega_0^2q^2)$, using $\Delta(t) = \Delta_0[1+\epsilon\cos(\omega t)]$ produces resonant forcing in the modified Hamiltonian

$$H' \approx \Omega_0 J - \frac{\Omega_0^2 \Delta_0 \epsilon}{4} J \sin(2\phi - \omega t),$$

leading to Mathieu-type instability when $$|\Omega_0-\omega/2| < (\Omega_0^2\Delta_0\epsilon)/4$$. This resonance causes orbits to experience exponential growth, despite measure preservation.

• Phase-space-dependent step, $\Delta = \Delta(q,p)$: Here, changing the time variable via $dt = \Delta(q,p) d\tau$ results in equations not of canonical Hamiltonian form. Standard symplectic integrators, which require canonical structure, then fail to preserve invariants—yielding secular energy or angular momentum drift and breakdown of classical phase-space volume conservation with respect to $dq \wedge dp$.

2. Symplectic Regularization Methodologies

To eliminate or regularize these pathologies, two structural approaches are developed:

Extended Phase Space Method

• Embedding: The time variable is introduced as $q_0 = t$ with conjugate momentum $p_0$, and one constructs the canonical extended Hamiltonian

$$K(q, p, q_0, p_0) = \frac{H(q, p, q_0) + p_0}{\rho(q, p)},$$

where $\rho(q, p) \propto 1/\Delta(q, p)$.

• Canonicalization: In the extended $(q, p, q_0, p_0)$ phase space, the evolution regains canonical form. Any standard, fixed-step symplectic integrator (e.g., leapfrog, Crank–Nicolson) applied to this extended system with step $h$ in $\tau$ preserves the required invariants.
• Numerical Implications: Surface-of-section and energy error diagnostics confirm boundedness of invariants and absence of unphysical resonant islands—even for spatially-variable time steps that would otherwise induce instabilities.

Non-Canonical Mixed-Variable Generating Function Method

• Non-canonical structure: For 2D phase space, one defines a density $\rho(x, y)$, leading to conservation of the two-form

$\rho(x, y) \, dx \wedge dy.$

• Mapping via generating function: Introducing a mixed-variable generating function $F(x, Y)$, the mapping

$$(x, y) = (x, Y) - h v(x, Y),~~~~ (X, Y) = (x, Y) + h u(x, Y),$$

with $u(x, y) = \partial H / \partial y$, $v(x, y) = -\partial H / \partial x$, and composing with its transpose map (via function $G$), produces the "non-canonical symmetrized leapfrog" (NSL) integrator.

• Structure preservation: This procedure is second-order accurate and rigorously preserves the non-canonical bracket, ensuring conservation of $\int \rho(x, y)\, dx\, dy$ and robust long-term boundedness of invariants.

Comparative Features

Method	System Size	Canonical Form	Error Boundedness
Extended Phase Space	Increases $(n+1)$	Restores canonical	Bounded
NSL (non-canonical generating func.)	2D	Non-canonical	Bounded

In higher dimensions, only the extended phase-space method generalizes directly. The non-canonical approach is tailored for 1 DOF.

3. Numerical Evidence of Regularization

The efficacy of these methods is demonstrated via numerical experiments:

• Time-dependent step, $\Delta(t)$: In the presence of resonant conditions, unregularized integrators show spiraling orbits and exponential growth in Hamiltonian; Poincaré sections reveal the emergence of spurious islands.
• Phase-space-dependent step, $\Delta(q,p)$: Direct variable-step integration leads to secular (linear) growth of energy error. Both extended phase space and NSL regularized integrators avoid secular drift, maintain bounded energy error, and do not introduce artificial resonances.
• Performance Summary: The regularization step suppresses parametric instabilities and eliminates secular error growth. Both methods achieve symmetry, time-reversibility, and robust long-term conservation.

4. Optimizing the Regularization Step: Error Equidistribution

The time-step regularization is further refined via optimization based on backward error analysis:

• Local error scaling: For a given integrator,

$e = \Delta^2 w(t),$

where $w(t)$ is a function of the velocity field's spatial and temporal derivatives.

• Global error accumulation over $[0, T]$:

$$E = h^2 \int \left( \frac{dt}{d\tau} \right)^3 w(t) d\tau,$$

• Equidistribution principle: Minimizing $E$ leads to an optimal density for the time transformation,

$\rho_{\mathrm{opt}}(q, p) = C\, [w(t)]^{1/3},$

or, equivalently, an optimal time step,

$$\Delta_{\mathrm{opt}} \propto 1/\rho_{\mathrm{opt}}.$$

• Numerical validation: Comparison of uniform, arc-length, and optimally-adapted schemes (as above) shows the optimal step—as determined by backward error analysis—minimizes both local and global error; interpolating between these schemes yields a strict minimum for error at $\beta=1$ corresponding to $\Delta = \Delta_{\mathrm{opt}}$.

5. Analytical Formulations

Key formulas codifying the symplectic regularization step:

• Extended phase-space Hamiltonian: $$K(q, p, q_0, p_0) = [H(q, p, q_0) + p_0]/\rho(q, p)$$
• Parametric instability criterion: $$|\Omega_0-\omega/2| < (\Omega_0^2 \Delta_0 \varepsilon)/4$$
• Non-canonical preserved two-form: $\rho(x,y) dx \wedge dy = \rho(X,Y) dX \wedge dY$
• Symmetrized NSL mapping: $(x,y)\mapsto (X,Y)$ as per the mapping via $F$ and $G$
• Optimal density for error equidistribution: $\rho_{\mathrm{opt}} \sim [w(t)]^{1/3}$

6. Significance and Broader Applications

The symplectic regularization step, as constructed in (Richardson et al., 2011), is essential for robust long-term simulation of Hamiltonian systems when adaptive step-size selection is required. It ensures that geometric invariants are not destroyed by naively variable steps. The result is numerical schemes with error boundedness that scale optimally with step-size and no artificial secular growth or instability. The methodology is applicable in celestial mechanics, molecular dynamics, plasma physics, and any setting requiring geometric integration under adaptive error control. The theoretical framework—especially when combined with backward error analysis and optimal error equidistribution—provides guidelines for principled algorithm construction in geometric numerical integration.