Hamilton's principal function: Information from Answers.com

The Hamilton's principal function, $S$ , is defined by the Hamilton–Jacobi equation (HJE), another alternative formulation of classical mechanics. This function $S$ is related to the usual action, $\mathcal{S}$ , by fixing the initial time $t 1$ and endpoint $\mathbf{q}_{1}$ and allowing the upper limits $t 2$ and the second endpoint $\mathbf{q}_{2}$ to vary; these variables are the arguments of the function $S$ , (see). In other words, the action function, $S$ , is the indefinite integral of the Lagrangian with respect to time.
The Hamilton's principal function is also a generating function of canonical transformation which makes the transformed Hamiltonian, $K$ , to be identically zero

$K = H + \frac{\partial S}{\partial t} = 0.$

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 the simple harmonic oscillator. For the simple harmonic oscillator the Hamiltonian has the form

$H =\frac{p^{2}}{2m} + \frac{kq^{2}}{2}.$

Substitution of this into Hamilton–Jacobi equation

$H\left(q_{1},\dots,q_{N};\frac{\partial S}{\partial q_{1}},\dots,\frac{\partial S}{\partial q_{N}};t\right) + \frac{\partial S}{\partial t}=0$

yields

$\frac{1}{2m}\left(\frac{\partial S}{\partial q}\right)^2 + \frac{kq^{2}}{2} + \frac{\partial S}{\partial t}=0.$

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

$S(q, E, t) = W(q) - E\cdot t$

where the time-independent function $W(\mathbf{q})$ is called the Hamilton's characteristic function and $E$ is a constant which turns out to be the energy.
If we substitute this expression back into above equation, we get

$\left(\frac{d W}{d q}\right)^2= 2mE - mkq^2$

where the partial derivative has been replaced by total derivative since $W(\mathbf{q})$ is the function of only one variable.
Finally for $W(\mathbf{q})$ we get

$W(q)= \int\sqrt{2mE - mkq^2}\,dq.$

Therefore the Hamilton's principal function for the simple harmonic oscillator is

$S(q, E, t)= \int\sqrt{2mE - mkq^2}\,dq -E\cdot t.$

Example

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 $q (t)$ and the momentum $p (t)$ . 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 $S (q, E, t)$ the constant $E$ plays the role of new generalized momenta. Since $q_{k} = \frac{\partial S}{\partial p_{k}}$ , the new constant generalized coordinate, denoted $β$ , is the partial derivative of $S$ with respect to $E$ .

$\beta=\frac{\partial S}{\partial E} = \int \frac{m}{\sqrt{2mE - mkq^2}}\,dq - t = \sqrt{\frac{m}{k}}\sin^{-1}{\sqrt{\frac{k q^2}{2E}}}-t$

or if we "turn inside out," the physical coordinate

$q(t) = \sqrt{\frac{2E}{k}}\sin(\omega t + \varphi)$

where $\omega^2= \frac{k}{m}$ and $\varphi= \beta\sqrt{\frac{k}{m}}$ .
The physical momenta can be found using $p_{k} = \frac{\partial S}{\partial q_{k}}$ which gives

$p(t) = \sqrt{2mE}\cos(\omega t + \varphi)$ .

These results are the same with results which one would have obtained for the simple harmonic oscillator using other methods than Hamilton–Jacobi equation. And also, here we see that the constant $E$ is indeed the total energy of the simple harmonic oscillator.

References

H. Goldstein (2002). Classical Mechanics. Addison Wesley. ISBN 0-201-65702-3.
L. D. Landau and E. M. Lifshitz (2000). Mechanics. Butterworth Heinemann. ISBN 0-7506-2896-0.

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Hamilton's principal function

Example

See also

References