In physics, and more specifically in
Hamiltonian mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
, a generating function is, loosely, a function whose partial derivatives generate the differential equations that determine a system's dynamics. Common examples are the
partition function of
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a
canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations. This is sometimes known as ''form invariance''. Although Hamilton's equations are preserved, it need not ...
.
In canonical transformations
There are four basic generating functions, summarized by the following table:
Example
Sometimes a given Hamiltonian can be turned into one that looks like the
harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'':
\vec F = -k \vec x,
where ''k'' is a positive const ...
Hamiltonian, which is
For example, with the Hamiltonian
where is the generalized momentum and is the
generalized coordinate, a good canonical transformation to choose would be
This turns the Hamiltonian into
which is in the form of the harmonic oscillator Hamiltonian.
The generating function for this transformation is of the third kind,
To find explicitly, use the equation for its derivative from the table above,
and substitute the expression for from equation (), expressed in terms of and :
Integrating this with respect to results in an equation for the generating function of the transformation given by equation ():
To confirm that this is the correct generating function, verify that it matches ():
See also
*
Hamilton–Jacobi equation
In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...
*
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
References
{{Reflist
Classical mechanics
Hamiltonian mechanics