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Variational Calculus

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Variational Calculus, Hamilton and Lagrange mechanics
  • Variational Calculus
  • Hamilton mechanics
  • Motion around an equilibrium, linearization
  • Phase space, Liouville’s equation
  • Generating functions

Variational Calculus


Starting with:


the equations of Lagrange can be derived:


When there are additional conditions applying to the variational problem J(u)=0 = 0 of the type K(u) =constant, the new problem becomes:



Hamilton mechanics


The Lagrangian is given by: Lagrangian The Hamiltonian is given by: Hamiltonian
In 2 dimensions holds: Lagrangian.

If the used coordinates are canonical the Hamilton equations are the equations of motion for the system:
equations of motion for the system

Coordinates are canonical if the following holds: Coordinates are canonical where {,} is the Poisson bracket:
Poisson bracket

The Hamiltonian of a Harmonic oscillator is given by Hamiltonian of a Harmonic oscillator. With new coordinates coordinates obtained by the canonical transformation canonical transformation and , with inverse inverse and it follows: .

The Hamiltonian of a charged particle with charge q in an external electromagnetic field is given by:
Hamiltonian of a charged particle

This Hamiltonian can be derived from the Hamiltonian of a free particle Hamiltonian of a free particle with the transformations transformations and transformations . This is elegant from a relativistic point of view: this is equivalent to the transformation of the momentum 4-vector 4-vector. A gauge transformation on the potentials A gauge 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:
natural systems around equilibrium
natural systems around equilibrium


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:
phase space

If the equation of continuity, equation of continuity holds, this can be written as:

continuity
For an arbitrary quantity A holds:
arbitrary quantity A
Liouville’s theorem can than be written as:
Liouville theorem


Generating functions

Starting with the coordinate transformation:
coordinate transformation
one can derive the following Hamilton equations with the new Hamiltonian K:
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.


 
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