- Hamilton's principal function
The Hamilton's principal function is defined by the
Hamilton–Jacobi equation (HJE), another alternative formulation ofclassical mechanics . This function is related to the usual action, , by fixing the initial time and endpoint and allowing the upper limits and the second endpoint to vary; these variables are the arguments of the function , (see). In other words, the action function, , is the indefinite integral of the Lagrangian with respect to time.
The Hamilton's principal function is also a generating function ofcanonical transformation which makes transformedHamiltonian , , to be identically zero:
Hamilton's equations for transformed Hamiltonian imply that the new generalized coordinates and the new generalized momenta are constant.
As an explicit example let us construct the Hamilton's principal function for thesimple harmonic oscillator .For the simple harmonic oscillator the Hamiltonian has the form:
Substitution of this into
Hamilton–Jacobi equation :
yields
:
This can be solved by additive separation of variables. Since the Hamiltonian does not depend on time explicitly, we seek a solution in the following form
:
where the time-independent function is called the Hamilton's characteristic function and is a constant which turns out to be the energy.
If we substitute this expression back into above equation, we get:where the
partial derivative has been replaced bytotal derivative since is the function of only one variable.
Finally for we get:
Therefore the Hamilton's principal function for the
simple harmonic oscillator is:
Applications
To illustrate usefulness of the Hamilton's principal function, let us solve the problem of simple harmonic oscillator discussed above. For this we need to find the position and the momentum . As above stated the Hamilton's principal function is a generating function of canonical transformation and therefore can be taken as the type 2 generating function, which is the function of only the old
generalized coordinates and the new generalized momenta. Thus above in expression for the constant plays the role of new generalized momenta. Since , the new "constant" generalized coordinate, denoted , is the partial derivative of with respect to .:
or if we "turn inside out," the physical coordinate
:
where and .
The physical momenta can be found using which gives:.
These results are the same with results which one would have obtained for the
simple harmonic oscillator using other methods thanHamilton–Jacobi equation . And also, here we see that the constant is indeed the total energy of the simple harmonic oscillator.ee also
*
Hamilton's equations
*Canonical transformation
* constants of motion
*Hamiltonian vector field
* In control theory, seeHamilton-Jacobi-Bellman equation .
*WKB approximation References
*
*cite book | author=L. D. Landau and E. M. Lifshitz | title=Mechanics | publisher=Butterworth Heinemann | year=2000 | id=ISBN 0-7506-2896-0
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