Mathematical & Computational Physics — 60 Terms Defined

Perturbation Theory Mathematical & Computational Physics Approximation Power Series A set of mathematical approximation methods for finding an approximate solution to a complex problem by starting from the exact solution of a simpler unperturbed problem. H = H₀ + λ H', E_n = E_n^(0) + λ E_n^(1) + λ² E_n^(2) + ...

Physical Simulation Mathematical & Computational Physics Computational Modeling The use of computer programs and numerical algorithms to simulate the time-dependent physical behavior of real-world or theoretical systems. x(t + Δt) = x(t) + v(t) Δt + ½ a(t) Δt²

Poisson Brackets Mathematical & Computational Physics Phase Space Operation An algebraic operation between two phase space functions describing time evolution and conservation laws in Hamiltonian mechanics. {f, g} = ∑ ((∂f / ∂q_i)(∂g / ∂p_i) - (∂f / ∂p_i)(∂g / ∂q_i))

Round-off Error Mathematical & Computational Physics Floating-Point Precision Error The difference between the exact mathematical value of a function and the value computed using finite-precision floating-point arithmetic. Machine Epsilon ε_mach ≈ 2⁻⁵³ ≈ 1.11 × 10⁻¹⁶ (IEEE 754 Double Precision)

Runge-Kutta Methods Mathematical & Computational Physics Numerical ODE Integration Method A family of iterative numerical integration algorithms used to solve ordinary differential equations with high accuracy. RK4: y_(n+1) = y_n + (h / 6) (k₁ + 2 k₂ + 2 k₃ + k₄)

Spherical Harmonics Mathematical & Computational Physics Spherical Surface Functions A set of special orthogonal functions defined on the 2D sphere surface serving as angular solutions to Laplace's equation in 3D spherical coordinates. Y_l,m(θ, φ) = √(((2l+1)/(4π)) ((l-m)!/(l+m)!)) P_l^m(cos θ) e^(i m φ)

Statistics and Probability Mathematical & Computational Physics Probability (0 to 1) The mathematical frameworks for analyzing random phenomena, quantifying measurement uncertainty, and modeling ensemble behavior in statistical mechanics. P(A|B) = P(B|A) P(A) / P(B), μ = E[X], σ² = Var(X)

Stokes' Theorem Mathematical & Computational Physics Vector Line/Surface Integral Equality A fundamental vector calculus theorem relating the surface integral of the curl of a vector field over an open surface S to the line integral of the field around its boundary curve ∂S. ∮_(∂S) F · dr = ∬_S (∇ × F) · dA

Taylor Series Mathematical & Computational Physics Infinite Polynomial Expansion An infinite polynomial series representation of a smooth function expressed in terms of its higher-order derivatives evaluated at a single point a. f(x) = ∑_(n=0)^(∞) (f^(n)(a) / n!) (x - a)^n

Tensor Mathematical & Computational Physics Geometric Multilinear Object A multilinear geometric object describing linear relationships between algebraic vectors, tensors, and scalars independent of coordinate choice. T'^ij = (∂x'^i / ∂x^k) (∂x'^j / ∂x^l) T^kl

Variational Principle Mathematical & Computational Physics Stationary Action Principle A fundamental physical principle asserting that true physical trajectories minimize or render stationary a scalar functional (such as action). δ S = δ ∫_t₁^t₂ L(q, q̇, t) dt = 0 (Hamilton's Principle)

Vector Calculus Mathematical & Computational Physics Differential Operators A branch of mathematics dealing with multivariable differentiation and integration of vector fields in 3D Euclidean space. ∇f (Grad), ∇ · F (Div), ∇ × F (Curl), ∇²f (Laplacian)