equationofmotion

An equation of motion is a mathematical relation that describes how a physical system evolves in time under given forces or interactions.

In classical mechanics, Newton's second law gives m d^2x/dt^2 = F(x,t) for a particle of mass m; the vector form F = ma yields second-order equations for many-body systems. Solutions require initial position and velocity.

The Lagrangian framework expresses dynamics via the Euler-Lagrange equations d/dt(∂L/∂qdot_i) - ∂L/∂q_i = 0, with generalized coordinates q_i and L = T - V. The Hamiltonian formalism recasts the dynamics in terms of q and conjugate momenta p, giving dq/dt = ∂H/∂p and dp/dt = -∂H/∂q.

In continuum physics, equations of motion are partial differential equations for fields, such as the wave equation, the Navier–Stokes equations, or the heat equation, expressing local conservation and constitutive relations.

In quantum mechanics, the time evolution of states is governed by the Schrödinger equation iħ ∂ψ/∂t = Hψ; in the Heisenberg picture, operators satisfy iħ dA/dt = [A,H], and Ehrenfest's theorem connects quantum motion to classical averages.

Common solution methods include analytic integration for simple systems and numerical schemes (for example, Runge–Kutta) for complex models. Equations of motion are central to physics, engineering, and applied sciences.