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Mathematical & Computational Physics

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Mathematical & Computational Physics

Hamiltonian Formalism

A formulation of classical mechanics describing system dynamics in 2N-dimensional phase space using position q_i and canonical momentum p_i.

H(q, p, t) = ∑ p_i q̇_i - L, q̇_i = ∂H / ∂p_i, ṗ_i = -∂H / ∂q_i Joules (Energy) Mathematical & Computational Physics

Hermite Polynomials

A sequence of classical orthogonal polynomials satisfying Hermite's differential equation y'' - 2x y' + 2n y = 0.

H_n(x) = (-1)^n e^(x²) (d^n / dx^n) (e^(-x²)) Dimensionless Orthogonal Polynomials Mathematical & Computational Physics

Hessian Matrix

A square matrix of second-order partial derivatives of a scalar-valued function, describing local surface curvature and local extrema…

H_ij = ∂²f / (∂x_i ∂x_j) Second Derivative Matrix Mathematical & Computational Physics

Hilbert Space

A complete abstract vector space possessing an inner product that allows length and angle to be defined for infinite-dimensional function…

⟨f, g⟩ = ∫ f*(x) g(x) dx, ||f|| = √(⟨f, f⟩) Complete Inner Product Space