Mathematical & Computational Physics

A change of coordinates in phase space (q, p) → (Q, P) that preserves the canonical form of Hamilton's equations.

(q, p) → (Q, P) such that Q̇_i = ∂K / ∂P_i, Ṗ_i = -∂K / ∂Q_i Phase Space Coordinate Transformation

The study of deterministic non-linear dynamical systems that exhibit extreme sensitivity to initial conditions (the butterfly effect).

Lyapunov Exponent: δx(t) ≈ δx₀ e^(λ t), λ > 0 Divergence Rate λ (s⁻¹)

Mathematical & Computational Physics

The branch of mathematical analysis studying functions of complex variables f(z) = u(x,y) + i v(x,y).

∮_C f(z) dz = 2 π i ∑ Res(f, z_k) (Residue Theorem) Complex Function Theory

Mathematical & Computational Physics

A vector differential operator measuring the local circulation or rotation density of a 3D vector field around a point.

curl F = ∇ × F = det|[î, ĵ, k̂], [∂/∂x, ∂/∂y, ∂/∂z], [F_x, F_y, F_z]| [Vector Field Unit] / m

Mathematical & Computational Physics

The mathematical construction of a function that best matches a set of experimental data points according to statistical criteria.

Minimizing χ² = ∑ ((y_i - f(x_i, θ)) / σ_i)² Goodness of Fit χ²