Mathematical & Computational Physics

A formulation of classical mechanics where system dynamics are derived from a single scalar function, the Lagrangian L(q, q̇, t).

L = T - V (Kinetic Energy minus Potential Energy) Joules (Action S in J·s)

An integral transform that maps a real-valued time-domain function f(t) into a complex frequency-domain function F(s).

L{f(t)} = F(s) = ∫₀^(∞) f(t) e^(-s t) dt s = σ + i ω (Complex Frequency Domain)

Mathematical & Computational Physics

A standard mathematical regression technique for determining the best-fitting curve or line to a set of data points by minimizing the sum…

Minimizing S = ∑_(i=1)^(N) (y_i - f(x_i, β))² Statistical Regression Method

Mathematical & Computational Physics

A sequence of orthogonal polynomials satisfying Legendre's differential equation (1-x²) y'' - 2x y' + n(n+1) y = 0.

P_n(x) = (1 / (2^n n!)) (d^n / dx^n) ((x² - 1)^n) Dimensionless Orthogonal Polynomials

Mathematical & Computational Physics

The mathematical discipline studying vector spaces, linear transformations, matrices, and systems of linear equations.

A x = b, v · w = ∑ v_i w_i Vector Spaces & Matrices