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Is the separation of variables technique to solve PDEs only useful when there's a Laplacian involved? 74.15.136.9 ( talk) 00:59, 28 September 2012 (UTC)
Ordinarily, a Hamiltonian is derived, and taking its partial derivatives yields Hamilton's equations of motion, , . Suppose that instead one has equations of motion, or possibly only numerical approximations of these, but not the Hamiltonian. Assuming that the Hamiltonian giving rise to these equation is time-dependent, how might one establish an integration problem, probably numerically, to find the evolution of the Hamiltonian over time? I'll add that an arbitrary time dependence could be given to any Hamiltonian by simply appending some function to it, so I'll further qualify that the Hamiltonian in question should not have any such spurious time-dependence.
Further, suppose one has equations of motion of the form and , but it is not obvious if the functions and are the same, and further not obvious that these are Hamiltonian equations of motion. If on applying the method that will doubtless be detailed above, one assumes that the two functions are the same, integrates, and finds that the function appears to be conserved over time, is this sound evidence that the equations are Hamiltonian and that the Hamiltonian is time-independent?-- Leon ( talk) 15:49, 28 September 2012 (UTC)
Mathematics desk | ||
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< September 27 | << Aug | September | Oct >> | September 29 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Is the separation of variables technique to solve PDEs only useful when there's a Laplacian involved? 74.15.136.9 ( talk) 00:59, 28 September 2012 (UTC)
Ordinarily, a Hamiltonian is derived, and taking its partial derivatives yields Hamilton's equations of motion, , . Suppose that instead one has equations of motion, or possibly only numerical approximations of these, but not the Hamiltonian. Assuming that the Hamiltonian giving rise to these equation is time-dependent, how might one establish an integration problem, probably numerically, to find the evolution of the Hamiltonian over time? I'll add that an arbitrary time dependence could be given to any Hamiltonian by simply appending some function to it, so I'll further qualify that the Hamiltonian in question should not have any such spurious time-dependence.
Further, suppose one has equations of motion of the form and , but it is not obvious if the functions and are the same, and further not obvious that these are Hamiltonian equations of motion. If on applying the method that will doubtless be detailed above, one assumes that the two functions are the same, integrates, and finds that the function appears to be conserved over time, is this sound evidence that the equations are Hamiltonian and that the Hamiltonian is time-independent?-- Leon ( talk) 15:49, 28 September 2012 (UTC)