Newtonian Mechanics

Motions concerning large (compared to an atom) and slow (compared to the speed of light in vacuum) objects moving in Euclidean space.

Newton's theory of motion has much in common with an axiomatic system like Euclidian geometry, except that on a physical theory, even if the prediction has been shown to follow from the axioms, it is upon the experiment to verify its validity. Like an axiomatic system, Newtonian Mechanics starts from a set of terms that are undefined within the system : position, mass, time, force. It is assumed that we understand what these terms mean and that they are measurable quantities that correspond to properties of an object on the physical world. It is also assumed that the position in space, distance in space and position as a function of time is correctly described by Euclidian vector algebra. The next component of the axiomatic system is a set of axioms.

Info

Axioms : A set of unproven propositions involving the undefined terms, from which all other propositions in the system can be derived via logic and mathematical analysis.

Newton’s laws of motion, as written in Principia:

Newton's 1st Law:
Every body continues in its state of rest, or uniform motion in a straight-line, unless compelled to change that state by forces impressed upon it.
Newton's 2nd Law:
The change of motion (i.e., momentum) of an object is proportional to the force impressed upon it, and is made in the direction of the straight-line in which the force is impressed.
Newton's 3rd Law:
To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts.

Isolated system

No external force or torque acting on the system

Conservation of Momentum

Lets suppose N objects in a system. The total forces exerted on the $i^{t h}$ object are the summation of all forces $f_{i j}$ excluding i=j (Point 1: An object cannot exert force on itself). From Newton's 2nd Law:

m_{i} \cdot \frac{d^{2} \vec{r_{i}}}{d t^{2}} = \sum_{j = 1, j \neq i}^{N} \vec{f_{i j}}

The summation over all objects will be:

\sum_{j, i = 1, j \neq i}^{N} \vec{f_{i j}} = \sum_{i = 1}^{N} m_{i} \cdot \frac{d^{2} \vec{r_{i}}}{d t^{2}} = \sum_{i = 1}^{N} m_{i} \frac{d^{2}}{d t^{2}} \frac{\sum_{i = 1}^{N} m_{i} \cdot \vec{r_{i}}}{\sum_{i = 1}^{N} m_{i}} = M \frac{d^{2}}{d t^{2}} {\vec{r}}_{c m}

where M is the total mass of the system and $r_{c m}$ is the vector displacement of the center of mass of the system - an imaginary point whose coordinates are the mass weighted averages of the coordinates of the objects that constitute the object

{\vec{r}}_{c m} = \frac{\sum_{i = 1}^{N} m_{i} \cdot \vec{r_{i}}}{\sum_{i = 1}^{N} m_{i}}

From Newton's 3rd Law every $f_{i j}$ corresponds to a $f_{j i}$ which is equal an opposite and will cancel out:

M \frac{d^{2}}{d t^{2}} {\vec{r}}_{c m} = 0

Meaning that the center of mass of the system moves in a straight line with constant velocity irrespective of the forces acting between various components of the system (Newton's 1st Law).
The velocity of the center of mass of an isolated system is a constant of motion. Momentum $p = m * u = m \frac{d r}{d t}$ is therefore also a constant of motion and its is conserved.

Conclusion

The total momentum of an isolated system is a conserved quantity, irrespective of the nature of the forces acting between various components of the system.

Conservation of Angular Momentum

We can build the angular momentum of the ith object by taking the vector product of the position vector $r_{i}$ and Newton's 2nd Law summed for all the forces exerted on the ith object

m_{i} {\vec{r}}_{i} \times \frac{d^{2} \vec{r_{i}}}{d t^{2}} = \sum_{j = 1, j \neq i}^{N} \vec{r_{i}} \times \vec{f_{i j}}

\frac{d}{d t} (m_{i} \vec{r_{i}} \times \frac{d \vec{r_{i}}}{d t}) = \frac{d}{d t} L_{i}

The total angular momentum will be

L = \sum_{i = 1}^{N} L_{i}

Therefore:

\frac{d L}{d t} = \sum_{j, i = 1, j \neq i}^{N} \vec{r_{i}} \times \vec{f_{i j}}

From Newton's 3rd Law for every $r_{i}, f_{i j}$ the indices can be swapped and the matched pair can be rewritten:

r_{i} \times f_{i j} + r_{j} \times f_{j i} = (r_{i} - r_{j}) \times f_{i j}

Lets assume central forces acting between the various components (gravity)
$r_{i} - r_{j}$ will be a vector that connects the two objects and is therefore parallel to the force vector.
The vector product of the central force and the subtracted vectors will be zero.

\frac{d L}{d t} = 0

Conclusion

The total angular momentum of an isolated system is a conserved quantity, provided that the different components of the system interact via central forces.

Faq

How can the forces cancel out in non central forces?
What is an example of an isolated system with non-central forces that does not conserve anglular momentum?

Non-Isolated System

Linear momentum

Total force (internal + external) acting on the ith object:

f_{i} = \sum_{j = 1, j \neq i}^{N} f_{i j} + F_{i}

Summation over all objects:

\sum_{i = 1}^{N} m_{i} \frac{d^{2} r_{i}}{d t^{2}} = \sum_{i = 1}^{N} F_{i}

\frac{d P}{d t} = F

The center of mass of a system consisting of many point objects has analogous dynamics to a single point object, whose mass is the total system mass, moving under the action of the net external force.

Angular momentum

With the same process we derive:

\frac{d \vec{L}}{d t} = \vec{T}

where T is the net external torque

\vec{T} = \sum_{i = 1}^{N} {\vec{r}}_{i} \times {\vec{F}}_{i}

If there is a net external torque acting on the system then the total angular momentum evolves in time according to this equation and is unaffected by internal torques (proved only for central forces).