# 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, $x\left(b\right)$
• A set of constraints,
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:
• $\stackrel{˙}{x}\left(t\right)=f\left(x\left(b\right),b,t\right)$
• $\varphi \left(x\left(b\right),b,t\right)=0$
The initial conditions for the mechanical system are prescribed as:(1)
$x\left({t}_{0}\right)={x}_{0}\left(b\right)$

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