Starting with:
the equations of Lagrange can be derived:
When there are additional conditions applying to the variational problem
= 0 of the type K(u) =constant, the new problem becomes:
Hamilton mechanics
The Lagrangian is given by:
The Hamiltonian is given by: 
In 2 dimensions holds:
.If the used coordinates are canonical the Hamilton equations are the equations of motion for the system:
Coordinates are canonical if the following holds:
where {,} is the Poisson bracket:
The Hamiltonian of a Harmonic oscillator is given by
. With new coordinates 
obtained by the canonical transformation 
and 
, with inverse 
and 
it follows: 
.The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by:
This Hamiltonian can be derived from the Hamiltonian of a free particle
with the transformations 
and 
. This is elegant from a relativistic point of view: this is equivalent to the transformation of the momentum 4-vector 
. A gauge transformation on the potentials 
corresponds with a canonical transformation, which make the Hamilton equations the equations of motion for the system. Motion around an equilibrium, linearization
For natural systems around equilibrium the following equations are valid:
With
one receives the set of equations 
is substituted, this set of equations has solutions if 
= 0.This leads to the eigenfrequencies of the problem:
If the equilibrium is stable holds:
that 
.The general solution is a superposition if eigenvibrations.
Phase space, Liouville's equation
In phase space holds:
If the equation of continuity,
holds, this can be written as:
For an arbitrary quantity A holds:
Liouville's theorem can than be written as:
Generating functions
Starting with the coordinate transformation:
one can derive the following Hamilton equations with the new Hamiltonian K:
Now, a distinction between 4 cases can be made:
1.
, the coordinates follow from:
; 
; 
2.
, the coordinates follow from:
; 
; 
3.
, the coordinates follow from:
; 
; 
4.
, the coordinates follow from:
; 
; 
The functions F1, F2, F3 and F4 are called generating functions.