# Variational Calculus

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.