The Hamiltonian function (also Hamiltonian, after William Rowan Hamilton ) of a system of particles is, if there are no rheonomic (i.e. time-dependent) constraints , the total energy as a function of the locations and momenta of the particles and possibly time. It is a Legendre transform of the Lagrange function of the system. Instead of using the location and pulse coordinates , the functional relationship can also be expressed using the generalized location coordinates and generalized pulse coordinates.
The Hamilton function is defined by
and depends on
- time ,
- the generalized coordinates and
- the generalized impulses .
It is derived from a Legendre transformation of the Lagrange function with respect to the generalized velocities, which depends on the generalized coordinates and their velocities :
Here, on the right side with the speeds those functions
meant, which one gets when one takes the definition of generalized impulses
dissolves according to the speeds.
The total differential of the Hamilton function is:
Based on the product rule , one obtains
where, due to the definition of the generalized momentum, the first and last terms in the brackets have the sum 0, so that:
With the above notation of the total differential, the partial derivatives of the Hamilton function follow :
The total derivative of the Hamilton function with respect to time is identical to the partial:
So if the Hamilton function does not explicitly depend on time , its value is a conserved quantity :
The Hamilton function determines the temporal development of the particle locations and impulses using Hamilton's equations of motion :
The Hamilton operator also determines the time evolution in quantum mechanics . In many cases it is obtained from the Hamilton function by canonical quantization by reading the algebraic expression for as a function of operators and that satisfy the canonical commutation relations.
For a particle of mass that moves non-relativistically in a potential , the Hamilton function is composed of kinetic and potential energy:
For a relativistic, free particle with the energy-momentum relationship
applies to the Hamilton function
For the free relativistic particle with the Lagrangian function
the generalized momentum depends according to
on the speed. Conversely, the speed is therefore the function
of the momentum.
The Hamilton function of a one-dimensional harmonic oscillator is given by:
Charged particle in an electromagnetic field
In Cartesian coordinates ( ), the Lagrange function of a particle of charge moving through an electromagnetic field is
Here is the electrical potential and the vector potential of the magnetic field. The canonical impulse is
This equation can be rearranged so that the speed is expressed in terms of momentum:
If the expression for and is used in the definition of the Hamilton function, this results in:
- Herbert Goldstein, Charles P. Poole, Jr., John L. Safko: Classical Mechanics . 3. Edition. Wiley-VCH, Weinheim 2006, ISBN 3-527-40589-5 .
- Wolfgang Nolting: Basic Course Theoretical Physics 2. Analytical Mechanics . 7th edition. Springer, Heidelberg 2006, ISBN 3-540-30660-9 .