### Energy

Energy is a quantity that is constant, that is it does not change with time, for any closed system.

Energy is a scalar quantity, that is it is independent of direction.

Energy can exist in different forms and can sometimes be exchanged between these forms, examples of types of energy are:

#### Potential energy

Energy by virtue of being on a force field. for example gravity, electric and magnetic fields have the potential to do work.

#### Kinetic energy

Energy by virtue of motion, for a particle it is 0.5 * m * v^{2}

where:

- m = mass (scalar quantity)
- v = velocity (we can use the scalar modulus of velocity or if we use multivectors squaring a velocity vector will give a scalar quantity)

For a solid body it is more complicated because rotating solid objects also have kinetic energy, the kinetic energy of a rotating object can be calculated by summing the linear energy of all its particles. However we cannot get the total kinetic energy of a solid body just add the kinetic energy due to rotation to the kinetic energy due to linear motion. We must add another factor to give this equation:

T = 0.5 m v^{2} + 0.5 w^{t} [I] w + m(v•(w x r))

this is derived here.

#### Heat energy

At the molecular level this is the kinetic energy of the particles that make up the object, so this is a form of kinetic energy.

#### Wave energy

Such as sound, light, water waves. This is due to oscillation between other types of energy.

#### Chemical energy

This is a form of potential energy due to the molecular bonds.

#### Electrical energy

The kinetic energy of electrons in an electric field.

#### Mass

Since we are talking about classical mechanics on most of this site then we
usually assume that mass is constant. However, if we need to take into account
the effects of Einstein's special relativity then energy can be converted into mass
and mass into energy given by E=m c^{2}

### Work

One definition of energy is the ability to do work, where work is force times distance:

work = force • d displacement

where:

- • = dot product
- d = differential operator
- displacement = change in position vector
- force = force vector

I'm not sure this is a strictly valid definition, the second law of thermodynamics suggests that not all forms of energy can be turned into mechanical work?

### Momentum

Momentum is a quantity that is constant for any closed system (like energy).

Momentum is a vector quantity, that is, it has components in x,y and z dimensions (unlike energy) momentum in one dimension cannot be converted to momentum in another dimension, so momentum is conserved in x dimension and it is separately conserved in y and z dimensions.

Momentum cannot be converted to different forms (unlike energy) although momentum can be exchanged between different masses when they interact by means of forces or collisions. For any closed system the total momentum is always constant.

### Angular Momentum

Angular momentum is constant for any closed system (provided that the angular momentum of every object in the system is measured about the same fixed point).

Angular momentum is a vector quantity, it has components about the x,y and z axes (any mutually perpendicular axes) momentum in one dimension cannot be converted to momentum in another dimension, so momentum is conserved in x dimension and it is separately conserved in y and z dimensions.

Angular Momentum is conserved independently of linear momentum. If a force exists between two objects or the objects collide then both angular momentum and linear momentum may be transferred between the objects but neither the angular momentum or the linear momentum of the total system will change. See these examples.