Phase-Space Warps

📐 Definition

A phase-space warp is a differentiable map $(x,k) \mapsto (X,K)$ on $\mathbb{R}^{2d}$ whose Jacobian $J$ satisfies the symplectic condition

J^{T} \Omega J = \Omega,\qquad \Omega = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}

Equivalently, it can be generated by a type-II generating function $S(x,K)$ via

k = \partial_x S,\qquad X = \partial_K S

Domain and Codomain

Defined on open subsets of $\mathbb{R}^{2d}$ where derivatives exist. Output lives in the same phase space with preserved symplectic form.

⚙️ Key Properties

Volume preserving ( $\det J = 1$ ) and energy-structure respecting for Hamiltonian systems. Composition of phase-space warps remains symplectic.

Linear symplectic maps include rotations and shear matrices with $A^{T} \Omega A = \Omega$ . Small deformations generated by $S$ yield $J = I + O(\varepsilon)$ symplectic up to first order.

🎯 Special Cases and Limits

Linear symplectic maps include rotations and shears.
Small generating-function perturbations produce near-identity symplectic maps.

Warp gradients supply the Jacobian; complex-analytic warps furnish conformal slices of such maps in $d=1$ ; hyperexponential factors can parameterize amplitude-like components within $S$ .

Usage in Oakfield

Oakfield’s current “warp” implementation is not a symplectic phase-space coordinate transform; it is a value-level deformation applied to fields:

Analytic warp changes field samples according to analytic profiles and guarded gradient sampling, and can operate in phase-preserving polar mode for complex fields.
Remainder-based measurement uses warped vs reference fields to quantify deformation effects, often pairing warps with filters and mixers in operator graphs.

If you need true Hamiltonian/symplectic phase-space warps, they are not implemented as dedicated operators in the current runtime.

Historical Foundations

📜 Hamiltonian Mechanics

Canonical transformations and generating functions are classical tools in Hamiltonian mechanics, preserving the symplectic structure that encodes conservation laws and geometric invariants.

🌍 Modern Perspective

Symplectic maps remain central in geometric numerical integration and phase-space modeling.

📚 References

Hairer, Lubich, Wanner, Geometric Numerical Integration
Arnold, Mathematical Methods of Classical Mechanics