“Generating function (physics)”的意思、由来-开放百科全书

词条

Generating function (physics)

释义

In canonical transformations
Example
See also
References

Generating functions which arise in Hamiltonian mechanics are quite different from generating functions in mathematics. In physics, 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:

Generating function	Its derivatives
	and
	and
	and
	and

Example

Sometimes a given Hamiltonian can be turned into one that looks like the harmonic oscillator Hamiltonian, which is

For example, with the Hamiltonian

where p is the generalized momentum and q is the generalized coordinate, a good canonical transformation to choose would be

{{NumBlk|:|

|{{EquationRef|1}}}}

This turns the Hamiltonian into

which is in the form of the harmonic oscillator Hamiltonian.

The generating function F for this transformation is of the third kind,

To find F explicitly, use the equation for its derivative from the table above,

and substitute the expression for P from equation ({{EquationNote|1}}), expressed in terms of p and Q:

Integrating this with respect to Q results in an equation for the generating function of the transformation given by equation ({{EquationNote|1}}):

To confirm that this is the correct generating function, verify that it matches ({{EquationNote|1}}):

References

{{cite book | author=Goldstein, Herbert | title=Classical Mechanics | publisher=Addison Wesley | year=2002 | isbn=978-0-201-65702-9}}

2 : Classical mechanics|Hamiltonian mechanics

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

In canonical transformations

Example

See also

References