We have expressed the dynamics equations in terms of differential equations, but these relationships can be expressed in terms of integral equations, for example:

#### Momentum Equations

F = ma

where a=d²x/dt²

but there is always an equivalent integral form of these equations.

v*m = momentum = ∫ F dt

This is a line integral in the time dimension which shows us that it is independent of the path through the space dimensions. Therefore the space dimensions are symmetrical. See Noethers theorem page.

#### Energy Equations

energy = d(m v)/dt

where v=dx/dt

energy(work) = ∫ F dx

in 3D energy(work) = ∫ F•dx

where •=dot product

This is a line integral in the space dimensions which shows us that it is independent of time. Therefore the time dimension is symmetrical. See Noethers theorem page.

#### Action

In classical mechanics the Newtonian equations of motion are equivalent to minimizing the action over the set of all paths.

In Quantum Mechanics (QM) all paths have a probability but paths with a lower action have a higher probability.

Here we are concerned with classical mechanics. The action of a particle (point element of matter) is determined by the Lagrangian L(x,v) which is a function of its position and its velocity.

So the motion of the particle can be determined by minimising the action which is the integral of this Lagrangian:

∫ L(x,v) dt

between t1 and t2

To calculate the classical equations of motion using these methods we make a small variation in the path of the particle keeping the endpoints fixed.

This leads to:

δL | = | d | δL |

δx | dt | δv |

Which is the Euler-Lagrange equation

m | d²x | = - | ∂V(x) |

dt² | ∂x |

Where:

V=potential energy (mgh)

## Noethers Theorem