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
