Generating function (physics)

In physics, and more specifically in Hamiltonian mechanics, 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, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a canonical transformation.

In physics, and more specifically in Hamiltonian mechanics, 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, the Hamiltonian, and the function which acts as a bridge between two sets of canonical variables when performing a canonical transformation. ==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 Hamiltonian, which is H = aP^2 + bQ^2. For example, with the Hamiltonian H = \frac{1}{2q^2} + \frac{p^2 q^4}{2}, where is the generalized momentum and is the generalized coordinate, a good canonical transformation to choose would be This turns the Hamiltonian into H = \frac{Q^2}{2} + \frac{P^2}{2}, which is in the form of the harmonic oscillator Hamiltonian. The generating function for this transformation is of the third kind, F = F_3(p,Q). To find explicitly, use the equation for its derivative from the table above, P = - \frac{\partial F_3}{\partial Q}, and substitute the expression for from equation ( ), expressed in terms of and : \frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q} 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 ( ): q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q} ==See also== Hamilton–Jacobi equation Poisson bracket ==References==

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