Time (in s).

A time step (in s).

Velocity of a particle (in m/s).

A simple one-particle state, to get started quickly with mechanics of one particle.

An acceleration function gives the particle's acceleration as a function of the particle's state. The specification of this function is what makes one single-particle mechanics problem different from another. In order to write this function, add all of the forces that act on the particle, and divide this net force by the particle's mass. (Newton's second law).

Time derivative of state for a single particle with a constant mass.

Single Runge-Kutta step

The state of a single particle is given by the position of the particle and the velocity of the particle.

The associated vector space for the state of a single particle.

The state of a system of one particle is given by the current time, the position of the particle, and the velocity of the particle. Including time in the state like this allows us to have time-dependent forces.

An acceleration function gives the particle's acceleration as a function of the particle's state.

Time derivative of state for a single particle with a constant mass.

Single Runge-Kutta step

List of system states

The state of a system of two particles is given by the current time, the position and velocity of particle 1, and the position and velocity of particle 2.

An acceleration function gives a pair of accelerations (one for particle 1, one for particle 2) as a function of the system's state.

Time derivative of state for two particles with constant mass.

Single Runge-Kutta step for two-particle system

The state of a system of many particles is given by the current time and a list of one-particle states.

An acceleration function gives a list of accelerations (one for each particle) as a function of the system's state.

Time derivative of state for many particles with constant mass.

Single Runge-Kutta step for many-particle system