koordinaattiacceleration

koordinaattiacceleration, or coordinate acceleration, is a term used to describe the second time derivative of generalized coordinates in a chosen coordinate chart for a physical system. If a system is described by generalized coordinates q^i(t) with i = 1,...,n, then the velocity components are q̇^i and the coordinate accelerations are a^i = d^2 q^i/dt^2 = q̈^i.

In Cartesian coordinates on a flat space, the coordinate accelerations a^i correspond directly to the physical acceleration components of the particle. In curvilinear or non-Cartesian coordinates, however, the basis vectors depend on position, so the physical acceleration includes additional terms arising from the coordinate geometry. A common way to relate these terms uses connection coefficients Γ^i_jk associated with the coordinate chart: a^i = q̈^i + Γ^i_jk q̇^j q̇^k. Thus coordinate acceleration refers to the time evolution of the coordinates themselves, while the physical acceleration can differ in non-Cartesian frames.

Coordinate accelerations play a central role in dynamics formulated with generalized coordinates. In Lagrangian mechanics, the Euler–Lagrange equations involve q̈^i and determine the evolution of the coordinates given the system’s forces and constraints.

Example: in polar coordinates (r, θ), the coordinate accelerations are q̈^r = r̈ and q̈^θ = θ̈, while the physical acceleration components in the orthonormal basis are a_r = r̈ − r θ̇^2 and a_θ = r θ̈ + 2 ṙ θ̇.

See also: acceleration, generalized coordinates, curvilinear coordinates, Christoffel symbols, Euler–Lagrange equations.