Mathematical & Computational Physics
5 terms
Canonical Transformation
Mathematical & Computational Physics
Phase Space Coordinate Transformation
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
Chaos Theory
Mathematical & Computational Physics
Divergence Rate λ (s⁻¹)
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
Complex Analysis
Mathematical & Computational Physics
Complex Function Theory
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)
Curl
Mathematical & Computational Physics
[Vector Field Unit] / m
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]|
Curve Fitting
Mathematical & Computational Physics
Goodness of Fit χ²
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