- #1
Zorba
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Regarding the Hamilton-Jacobi equation in it's usual form, I am having trouble understanding the following statement from Goldstein they say
"When the Hamiltonian does not depend explicitly upon the time, Hamilton's principal function can be written in the form
[tex]S(q,\alpha,t)=W(q,\alpha)-at[/tex]
where [tex]W(q,\alpha)[/tex] is called Hamilton's characteristic function."
So why is this? I don't understand why it is required that there be no explicit dependence on the time, it's seems to be as though we should be able to do this anyways due to the form of the Hamilton-Jacobi equation...
"When the Hamiltonian does not depend explicitly upon the time, Hamilton's principal function can be written in the form
[tex]S(q,\alpha,t)=W(q,\alpha)-at[/tex]
where [tex]W(q,\alpha)[/tex] is called Hamilton's characteristic function."
So why is this? I don't understand why it is required that there be no explicit dependence on the time, it's seems to be as though we should be able to do this anyways due to the form of the Hamilton-Jacobi equation...