# 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, $\varphi (x(b),\text{}b,\text{}t)=0$

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:

- $\dot{x}(t)=f\left(x(b),b,t\right)$
- $\varphi (x(b),b,t)=0$

The initial conditions for the mechanical system are prescribed as:

$$x({t}_{0})={x}_{0}(b)$$

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.