Equations of Motion

The following notes are from Equations of Motion up to Dynamic Equations of Motion

Equations of Motion

Equations of motion describe the behaviour of a physical system as a set of mathematical functions in terms of dynamic variables, usually spatial coordinates and time, but may include momentum.

If the dynamics ofa system is known, the equations of motion are the solutions for the differential equations describing the motion of the dynamics

Equation of Momentum

$$p = mv$$

• momentum is $p$;
• mass is $m$;
• velocity is $v$;

Mathematical functions of motion are defined in a Euclidean Space.

Types

There are two main descriptions of motion: dynamics and kinematics

General planar motion

These are the kinematic equations for a particle traversing a path in a plane, described by $r=r(t)$ They are simply the time derivative of the position vector in plane polar coordinates using the definitions of physical quantities for angular velocity $\omega$ and angular acceleration $\alpha$. These are instantaneous quantities which change with time.

The position of the particle is $r = r(r(t),\theta(t)) = r \hat e _{r}$ ; where $\hat e _{r}$ and $\hat e _{\theta}$ are the polar unit vectors.