Nonlinear DAE
Nonlinear DAEs from the Principle of Stationary Action
Many physical systems are governed by constraints—relationships among variables that must be satisfied at all times. When such systems are described using variational principles, particularly the principle of stationary action, the resulting equations of motion often take the form of nonlinear differential-algebraic equations (DAEs).
Unlike standard ordinary differential equations (ODEs), DAEs involve both differential terms and algebraic constraints. These constraints may arise naturally from the system’s geometry, conservation laws, or imposed physical restrictions (such as rigid connections or incompressibility).
In this chapter, we explore how DAEs emerge from the variational formulation of mechanical systems with constraints. We begin by applying the stationary action principle to systems with holonomic and non-holonomic constraints, and show how Lagrange multipliers are introduced to enforce them. The resulting equations form a coupled system of differential and algebraic components—nonlinear in general—whose analysis and numerical solution require specialized techniques.
Our focus will be on understanding the structure of these equations, their geometric interpretation, and the challenges they pose for simulation and control.
Learning objectives
After this part, you should be able to derive constrained Euler–Lagrange equations, interpret the role of Lagrange multipliers, and explain why constraint differentiation can create drift in a numerical simulation.