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Newtonian Mechanics

Table of Contents​

The Very Basics​

Newton's Second Law​

The starting quantity we define is the position vector , a vector in that represents the position of a particle. The velocity of the particle is defined as

and the acceleration as

The inertial mass of the particle is defined in Newtonian mechanics as its resistance to an applied force. However, all experiments have shown that the inertial mass is equivalent to the gravitational mass ; thus, we will treat both masses as identical and denote them as . The momentum is thus

Newton's second law of motion is essentially a definition for the force;

and if is constant,

This equation applies in an inertial reference frame (one that is not accelerating). Had the frame be non-inertial, fictitious forces would exist and need to be accounted for in the law.

Rotational Quantities​

It is often useful to define analogous quantities that describe rotations rather than linear translations. Around a point (the origin of rotation), the angular momentum is

and the torque as

For rotations, Newton's second law has a close analogue that can be derived as follows:

Energy​

The work done by a force on a particle along a path is defined as

If the particle has a constant mass, this can be evaluated as

where is the kinetic energy defined as

Conservative Forces​

A force is conservative if any of the following equivalent conditions apply:

  • The force does not change a particle's total energy .
  • The force field can be expressed as the negative gradient of a scalar field (the potential), .
  • The work done by the force is path-independent.
  • The work done by the force along a closed loop is zero.

It is easy to prove that conservative forces conserve energy; the work done by the force is

Considering that ,

so

The other conditions come from vector calculus, which I have thoroughly delineated here.

Systems of Particles​

When we consider a system of particles, we need to distinguish internal forces and external forces. Internal forces are forces acting on a particle by another particle in the system. External forces are forces acting on a particle by a source outside the system.

We can denote the internal force from particle to particle as . Then, the total internal force acting on the -th particle is

Combining this with the external force , the second law, when applied to the -th particle, is

Newton's third law states that the forces exerted on two particles by each other have equal magnitude but opposite directions. In other words, . If we sum Equation for all , we get

By the third law, the terms cancel out for each pair of cancels out. Therefore we are left with

We shall just denote the total external force (on the left-hand side) as . On the right-hand side, we will introduce a new quantity called the center of mass , defined as the weighted average of the position vectors based on their masses;

Then the equation becomes

Similarly, the sum of the linear momentums gives

Therefore,

We can make a similar deduction for the rotational case; the total torque is

We can also distinguish between external torques and internal torques the same way;

The key part is that we can consider a pair of summed elements on the rightmost term;

If we assume that the separation vector has the same direction as the force, then the term simply vanishes. Then Equation simplifies to

And since ,

Note that in our two derivations we have assumed that the third law applies in two ways. First, that the forces are equal and opposite is the weak law of action and reaction. Second, that the forces' directions are parallel to their separation vector is the strong law of action and reaction. While they hold for most forces, some forces, such as electromagnetic ones, violate these assumptions.

We can construct a table to summarize the analogy between translations and rotations, listed below.

TranslationsRotations