Differential Equations of Motion of a Particle

Differential equations of motion connect particle kinematics with the forces acting on the particle. For a particle of constant mass, they follow directly from Newton's second law.

Vector equation

The fundamental equation of motion is:

$$m\frac{d^2\vec r}{dt^2}=\sum_i\vec F_i.$$

If forces depend on position, velocity, or time, the right-hand side may be written as $\vec F(\vec r,\vec v,t)$.

Cartesian equations

Projection onto fixed coordinate axes gives:

$$m\ddot x=\sum F_x,\qquad m\ddot y=\sum F_y,\qquad m\ddot z=\sum F_z.$$

This is a system of second-order differential equations for the particle coordinates.

Natural coordinates

For motion along a known path, projection onto the tangent and principal normal is often convenient:

$$m\frac{dv}{dt}=\sum F_\tau,\qquad m\frac{v^2}{\rho}=\sum F_n,$$

where $\rho$ is the radius of curvature of the path.

Initial conditions

To determine the motion uniquely, initial position and velocity are normally specified, for example $x(t_0)=x_0$ and $\dot x(t_0)=v_{0x}$. The integration constants are found from these conditions.

Choosing coordinates

The coordinate system should match the geometry of motion and force directions. One axis is enough for rectilinear motion; natural coordinates are often efficient for motion along a curved path.

Example

For vertical free fall without air resistance, with the $y$ axis directed upward, $m\ddot y=-mg$, hence $\ddot y=-g$. Integrating twice and applying the initial conditions gives the equation of motion.

Common mistakes

• confusing the sign of a force component with the force magnitude;
• omitting constraint reactions;
• integrating without applying initial conditions;
• treating $v^2/\rho$ as the total acceleration rather than its normal component.