1Newtonian dynamics of particles

IA Dynamics and Relativity

1.3 Newton’s Second Law

Newton’s second law is often written in the form of an equation.

Law. The equation of motion for a particle subject to a force F is

dp

dt

= F,

where

p

=

mv

=

m

˙

r

is the (linear) momentum of the particle. We say

m

is the

(inertial) mass of the particle, which is a measure of its reluctance to accelerate.

Usually, m is constant. Then

F = ma = m

¨

r.

Usually,

F

is specified as a function of

r,

˙

r

and

t

. Then we have a second-order

ordinary differential equation for r.

To determine the solution, we need to specify the initial values of

r

and

˙

r

, i.e.

the initial position and velocity. The trajectory of the particle is then uniquely

determined for all future (and past) times.