Gravitational or electrical forces are inverse squared laws:

Strikingly similar, they are both also conservative forces. For gravity:

and

for the electrical force. Defining the potentials , and , you get the gravitational and electrical potentials

Conservative fields are defined from these potentials

In general, for any field , if conservative, . The gravitational field reads, from newtonian gravity (module a sign)

and you would get in the coulombian field case! Focusing on the gravitational case (a similar field could be done with the electrical field)…The momentum

is conserved under any vertical (radial) gravitational field. Imagine you do a traslation

The momentum in the component reads ! Note the momentum in the or vertical component would not be conserved due to ! Thus, symmetry is important. Imagine a spring holding from the upper horizontal surface. Then

where and

with , then

and

Since the spring force is conservative, , the total energy is conserved. Note the symmetry that says does not depend on the time and it is constant!

Going 3D is important here. We will use components to avoid vector arrows for convenience. Newton’s second law is

If , then !

Kinetic energy for non-relativistic particles read

If is conserved, then the kinetic energy is also conserved. This is valid for the free particle. In the case of conservative forces, the potential energy reads

and it yields a uniform motion with solution

and

The first term is precisely:

And thus,

with

or

i. e., , Q.E.D. for any conservative force. holds for conservative forces with certain properties in the potential energy (depending on coordinates in a homogeneus way). For the harmonic oscillator:

and the solution

with at . in general, so

or any other sinusoidal waveform as well. The velocity

and then

as before!

Light can NOT be described with classical NEWTONIAN mechanics. It took several decades an roughly speaking several centuries to code electromagnetic laws into a single set of equations. Maxwell wrote the synthesis of our current electromagnetic knowledge of light:

- Gauss law for the electric field: . Equivalently, . For point particles, this law provides . Moreover, , and the gauge field . is the potential.
- Faraday’s law: , or equivalently .
- Gauss law for the magnetic field (no magnetic monopoles in standard electromagnetism): , or .
- Ampere’s law: . This original Ampere’s law does not conserve electric charge, so Maxwell added an extra term, the displacement current, yielding
.

The combination of the 4 equations above produces wave-like equations for :

Plane wave solutions are allowed, with or generally

Wave speed is given by

so Maxwell cleverly pointed out that light should be an electromagnetic wave! Furthermore, in general. Light can also be polarized. Polarization or fluctuations in the directions of is due to the transverse character of the electromagnetic waves. Maxwell’s equations unify into a single framework. All the electromagnetic phenomena from a common dynamics. Special relativity allows to condense Maxwell equations into and . Clifford algebra simplify these equations into a single . Differential forms also allows for such a simplification. Maxwell equations have a new invariance beyond galilean invariance: Lorentz invariance. Essentially, Maxwell equations imply Special Relativity.

In presence of matter, Maxwell equation must be completed with constitutive relations for the electromagnetic fields, plus

and the boundary conditions for

where are the polarization and the magnetization for the pair.

*Remark:* Beyond mechanics and light, today we care about entropic forces,

Entropic forces and conservative forces can be added

Only in the zero temperature limit, we get the usual conservative terms. The above force can be obtained from

i.e., the Helmholtz free energy.