Symplectic integrators and modified energy

In canonical coordinates, Hamilton's equations are $q'=H_p$ and $p'=-H_q$. A differentiable discrete map is symplectic when its derivative preserves the canonical skew form. In one degree of freedom this is equivalent to area preservation, but in higher dimensions determinant one is only necessary, not sufficient. The full matrix identity records how position and momentum directions pair. Symplecticity is a property of the update map, not a property inferred from an energy plot.

For a separable Hamiltonian with kinetic and potential terms, a kick in momentum followed by a drift in position gives symplectic Euler. Each substep is the exact flow of one Hamiltonian part, and their composition is symplectic when the step is fixed independently of state. Reversing the substep order gives a related but different first-order method. Symmetric compositions such as Störmer–Verlet can raise the order while preserving the same geometry, provided their actual substeps satisfy the stated structure.

For the unit oscillator, kick momentum by minus h times the old position, then drift position by h times the new momentum. The resulting two-by-two matrix has determinant one. Its trace is two minus h squared, so its eigenvalues lie on the unit circle when the positive step is strictly below two. At the boundary, the repeated eigenvalue can produce growing iterates; above it an unstable reciprocal pair appears. Symplecticity therefore does not remove step restrictions for a particular oscillatory problem.

Substitution into the displayed quadratic verifies an exact invariant for this linear method. That quadratic differs from the physical energy by a cross term proportional to h. It is positive definite for step magnitude below two, giving a direct bound on the oscillator iterates. Physical energy may oscillate rather than remain constant. This is a specific algebraic result for the oscillator, stronger and more precise than simply asserting that every symplectic method conserves energy.

Backward error analysis seeks a modified differential equation whose flow matches a numerical step order by order in h. For symplectic maps near a Hamiltonian flow, the modified vector field can be Hamiltonian under appropriate local assumptions. The resulting modified Hamiltonian is generally a formal asymptotic series, not a convergent expression that the method exactly preserves. Truncation, smoothness, bounded trajectories, and the size of the step relative to the dynamics determine what quantitative conclusions can be justified.

Long-time near-energy results require additional hypotheses and do not guarantee accurate phase forever. State-dependent adaptive step selection can also destroy the symplecticity of a method that is symplectic at fixed h. Prescribing a sequence of state-independent symplectic maps preserves their geometric composition, but a single fixed modified Hamiltonian need not survive that variation. For this lesson, the oscillator's exact quadratic invariant is the concrete guarantee; the general modified-equation picture explains a mechanism rather than supplying an unconditional conservation theorem.