Trimester 04 · Course detail

Numerical Methods & Analytical Mechanics

Variational and Hamiltonian mechanics, numerical analysis and computational linear algebra, probability theory and stochastic foundations, closed by the Associate capstone and adversarial viva voce

Credits: 15
Sequence: 4 of 8
Structure: 4 subjects
Status: Full syllabus

Curriculum focus

Variational and Hamiltonian mechanics, numerical analysis and computational linear algebra, probability theory and stochastic foundations, closed by the Associate capstone and adversarial viva voce

Evidence of mastery

Symplectic physics engine & non-linear solver repo + 45-minute live oral defense

Subject 01

Analytical Mechanics & Variational Principles

Official syllabus code: FUW 1100-3INZ11

Classical mechanics rebuilt on the calculus of variations: Lagrangian and Hamiltonian formulations, symplectic structure, and the Hamilton–Jacobi theory.

Variational Foundations

The Calculus of Variations & Hamilton's Principle

Derive the Euler–Lagrange equations from stationary-action functionals and formulate Hamilton's principle of stationary action as the foundation of dynamics.

Lagrangian Dynamics, Generalized Coordinates & Cyclic Invariants

Work in generalized coordinates, identify cyclic coordinates, and extract conserved momenta as the invariants tied to ignorable variables.

Hamiltonian Structure

Hamiltonian Mechanics, Legendre Duals, Canonical Equations & Routhians

Pass to the Hamiltonian via the Legendre transform, derive Hamilton's canonical equations, and form Routhians for mixed Lagrangian–Hamiltonian descriptions.

Symplectic Manifolds, Poisson Brackets & Integrability (Liouville–Arnold)

Treat phase space as a symplectic manifold, develop Poisson bracket algebra, and prove Liouville–Arnold integrability for systems with sufficient constants of motion.

Canonical Transformations, Generating Functions & Hamilton–Jacobi Theory

Characterize canonical transformations through generating functions and solve dynamics via the Hamilton–Jacobi equation, the gateway to wave-mechanical analogies.

Subject 02

Numerical Analysis & Computational Linear Algebra

Official syllabus code: MIMUW 1000-214bMN

Rigorous numerics for linear systems and approximation: conditioning and floating-point error, direct and iterative solvers, and quadrature.

Error, Conditioning & Direct Solvers

Matrix Conditioning, Floating-Point Error & Perturbation Analysis

Quantify sensitivity through condition numbers, model IEEE floating-point round-off, and bound solution perturbations for ill-conditioned systems.

Direct Solvers: LU, Cholesky, QR (Householder/Givens) & SVD

Implement and compare LU with pivoting, Cholesky factorization for SPD systems, Householder and Givens orthogonal reductions, and the SVD as the universal solver.

Iterative Methods & Approximation

Krylov Subspaces: Arnoldi, Lanczos, GMRES & Conjugate Gradient

Build Krylov subspace projections via Arnoldi and Lanczos, and analyze GMRES for nonsymmetric systems and CG for SPD systems with optimality and convergence bounds.

Approximation Theory: Chebyshev Polynomials, Minimax & Hermite Splines

Study Chebyshev polynomials and the minimax property, control Runge phenomena, and construct Hermite splines for interpolation with prescribed derivatives.

Numerical Quadrature: Gauss-Legendre, Gauss-Radau & Adaptive Stieltjes

Derive Gaussian quadrature from orthogonal polynomials, treat fixed-endpoint rules with Gauss–Radau nodes, and build adaptive schemes via Stieltjes procedures.

Subject 03

Probability Theory & Stochastic Foundations

Official syllabus code: MIMUW 1000-213bRP

Measure-theoretic probability: Kolmogorov's axioms, expectation as integration, conditional expectation, and the classical limit theorems.

Axiomatic Foundations

Kolmogorov Axioms, Probability Spaces (Ω, F, P) & Borel–Cantelli

Construct probability spaces over σ-algebras and apply the first and second Borel–Cantelli lemmas to almost-sure event sequences.

Random Variables, Cumulative Distributions & Pushforward Measures

Define random variables as measurable maps, characterize them by distribution functions, and transport measures through pushforwards.

Expectation, Conditioning & Limit Theorems

Expectation as Lebesgue Integral, Moments, Generating Functions & Inequalities

Define expectation as a Lebesgue integral, derive moment and generating-function machinery, and prove Markov, Chebyshev, and Jensen inequalities.

Independence & Conditional Expectation with respect to σ-Algebras

Formalize independence of σ-algebras and construct conditional expectation E[X | G] as an orthogonal projection in L², extended to L¹.

Limit Theorems: Weak/Strong Laws of Large Numbers & the Central Limit Theorem

Prove the weak and strong laws of large numbers and establish the CLT via characteristic functions, with convergence-in-distribution tools.

Subject 04

Associate Capstone & Adversarial Viva Voce

Official syllabus code: CAPS-201 / Registrar Gate

The 60-credit milestone gate: a comprehensive synthesis across Panels 1–4, a full engineering artifact, and a live adversarial oral defense before the AI Examination Tribunal.

Capstone Synthesis & Defense

Comprehensive Mathematical Synthesis across Panels 1–4 (60 Credits)

Integrate analysis, fields, dynamical systems, and probability from the first four trimesters in a unified written synthesis examination.

End-to-End Symplectic Physics Engine & Non-Linear Solver Repo

Design and ship a complete symplectic physics engine with non-linear solvers — the portfolio artifact evidencing the trimester's mechanics and numerics.

45-Minute Live Adversarial Oral Defense before the AI Examination Tribunal

Defend the synthesis and the engine in a live 45-minute viva voce before the AI Examination Tribunal, the registrar gate to the Associate Fellow Diploma.