Start & Prepare

Choose a goal, try a concrete example, and repair only the preparation needed for the next task.

Start with What makes a system integrable? if the subject is new to you. It uses a nonlinear oscillator to explain conservation and the scope of an integrability claim. For a concise technical comparison, use Definitions of integrability compared.

Choose a concrete problem

Four sequences are available. Each develops its own mathematical tools, includes exercises with solutions, and ends in a reproducible calculation. You can enter through any of them; mechanics is not a prerequisite for the quantum route, and the probability route requires no quantum mechanics.

Start from	Choose	Preparation used	Result you will establish
Moving particles	Open Toda	Differentiation and small matrices	Equations, conserved eigenvalues, independent commuting integrals and a tested trajectory
Quantum spin states	The XXX chain	Complex vectors and matrix multiplication	One- and two-magnon states, scattering and finite-ring Bethe equations
A wave that keeps its shape	KdV solitons	Chain rule, elementary ODEs and integration by parts	A travelling pulse, its spectral data, reconstruction and numerical convergence
Random particle hops	Finite-ring TASEP	Elementary probability and finite matrices	A Markov generator, exact stationary current and a tested Bethe decay mode

Try the entry calculation

Open the chosen sequence’s preparation check: Toda derivatives and matrices, spin-chain vectors and wave numbers, travelling profiles and boundary terms, or probability and transition rates. Attempt the calculation before opening its answer. If one step is unfamiliar, use that explanation and the relevant lesson’s local repair; you do not need to complete every subject in the chapter map first.

Each route can be read without installing software. Python and NumPy are needed to reproduce or change the numerical experiment. The project pages give the code, inputs, tested environment, expected results and the limits of each check.

If differential equations are unfamiliar, use Differential equations & phase portraits to check a proposed solution and its initial data, read the direction of motion, and recognize when a solution can cease to exist. Return from its small examples to the oscillator, Toda or travelling-wave lesson.

If Hamiltonian mechanics is unfamiliar, start with the bracket and canonical-coordinate bridge and then solve an oscillator two ways. These short preparations explain what conservation and canonical variables mean before the Toda calculations use them. The Liouville–Arnold reference states sufficient hypotheses for action–angle motion near a regular invariant torus. Practice applying them to pendulum motion and its separatrix.

For the quantum route, the linear algebra bridge develops complex inner products, eigenvector checks, tensor-product bases and the commutators that preserve a sector. Use it when a matrix calculation in the XXX entrance is unfamiliar, then return to that lesson.

Read for a purpose

Use Learn when you want a guided calculation and practice. Use the Library when you want the longer derivation, Models to establish the equation and regime, and Reference to resolve a convention or find a source. These pages support the same calculation from different directions.

Experienced readers can go directly to the Toda proof, two-magnon derivation, commuting-transfer-matrix proof, or KdV spectral construction, then use the exercises to check a less familiar step. The four-lesson algebraic follow-up develops the transfer construction and regular Bethe vectors with intermediate practice. The Practice & Projects page collects the available computations.

The chapter map below describes the broader preparation material planned for the site. Linked readings are available; plain-text titles are future coverage.