Equations of Motion for a Multibody System

This section describes equations of motion.

Consider a mechanical system defined in the following way:

A set of design variables, $b$
A set of states, $x (b)$
A set of constraints, $ϕ (x (b), b, t) = 0$

The set of states, x, consist of:

Displacements
Velocities
Lagrange Multipliers
Auxiliary variables such as user-variables and states for user-differential equations

With this notation, the equations of motion for a mechanical system can be expressed as:

$\dot{x} (t) = f (x (b), b, t)$
$ϕ (x (b), b, t) = 0$

The initial conditions for the mechanical system are prescribed as:(1)

x (t_{0}) = x_{0} (b)

For notational simplicity, henceforth, $x (b)$ will simply be referred to as $x$ . The states, $x$ , are required to satisfy the above equations at all points in time.