1Newtonian dynamics of particles

IA Dynamics and Relativity

1 Newtonian dynamics of particles

Newton’s equations describe the motion of a (point) particle.

Definition

(Particle)

.

A particle is an object of insignificant size, hence it can

be regarded as a point. It has a mass m > 0, and an electric charge q.

Its position at time

t

is described by its position vector,

r

(

t

) or

x

(

t

) with

respect to an origin O.

Depending on context, different things can be considered as particles. We

could consider an electron to be a point particle, even though it is more accurately

described by the laws of quantum mechanics than those of Newtonian mechanics.

If we are studying the orbit of planets, we can consider the Sun and the Earth

to be particles.

An important property of a particle is that it has no internal structure. It can

be completely described by its position, momentum, mass and electric charge.

For example, if we model the Earth as a particle, we will have to ignore its own

rotation, temperature etc.

If we want to actually describe a rotating object, we usually consider it as a

collection of point particles connected together, and apply Newton’s law to the

individual particles.

As mentioned above, the position of a particle is described by a position

vector. This requires us to pick an origin of the coordinate system, as well as an

orientation of the axes. Each choice is known as a frame of reference.

Definition

(Frame of reference)

.

A frame of reference is a choice of coordinate

axes for r.

We don’t impose many restrictions on the choice of coordinate axes. They

can be fixed, moving, rotating, or even accelerating.

Using the position vector

r

, we can define various interesting quantities which

describe the particle.

Definition (Velocity). The velocity of the particle is

v =

˙

r =

dr

dt

.

Definition (Acceleration). The acceleration of the particle is

a =

˙

v =

¨

r =

d

2

r

dt

2

.

Definition (Momentum). The momentum of a particle is

p = mv = m

˙

r.

m

is the inertial mass of the particle, and measures its reluctance to accelerate,

as described by Newton’s second law.