dynamics

Dynamics is the branch of physics and applied mathematics concerned with the motion of bodies and the evolution of systems over time. It studies the causes of motion—forces, torques, and energy transfers—and how systems change according to equations of motion. It is contrasted with kinematics, which describes motion without regard to its causes.

In classical mechanics, dynamics uses Newton's laws and, in more general form, Lagrangian and Hamiltonian formalisms to model rigid bodies, fluids, and deformable media. The central objects are differential equations that relate state variables such as position, velocity, and momentum to applied forces and constraints. Stability analysis, conservation laws, and symmetries play important roles.

Specialized areas include fluid dynamics, which concerns the motion of liquids and gases (often described by the Navier–Stokes equations); celestial dynamics, which studies planetary and satellite motion under gravity; and rigid- and soft-body dynamics, which address how bodies move and deform under loads.

Dynamical systems theory abstracts these ideas to mathematical models that evolve in time, either continuously or discretely. Key concepts include state space, trajectories, fixed points, stability, bifurcations, and chaotic behavior. This framework underpins much of modern applied mathematics and theoretical physics.

In engineering and control, dynamics analyzes the response of structures, machines, and vehicles to time-varying inputs, often focusing on vibrations and transient behavior. In biology, population dynamics models the growth and interaction of species; in economics and epidemiology, dynamic models track evolving variables over time.

In music and acoustics, dynamics denotes the relative loudness of sound and its expressive changes. Computational dynamics uses numerical methods to simulate evolving systems, employing time-stepping schemes and stability considerations to approximate real-world behavior.