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September 28

Separation of Variables

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)[reply]

See the examples given in separation of variables. Ssscienccce (talk) 08:47, 28 September 2012 (UTC)[reply]

All the examples therein involve the Laplacian. I want to know if this is an accident or if separation of variables is used only for equations involving a Laplacian. 74.15.136.9 (talk) 16:22, 28 September 2012 (UTC)[reply]

You can use separation of variables at other times. For example, replace the second-order partial derivatives in those examples with first-order partials, or even third-order partials, etc. Separation of variable will work just as well for those cases. Laplacians are common because they occur in many physically motivated circumstances, but they aren't necessary in order for separation of variables to work. Dragons flight (talk) 20:54, 28 September 2012 (UTC)[reply]

Now that this question is being asked here, I have always wondered what exactly is the criteria for separation of variables to work. Someone once told me that as long as the underlying coordinate system is orthogonal, separation of variables will work. Is that true?174.16.229.18 (talk) 23:32, 28 September 2012 (UTC)[reply]

A short partial answer: If the equation is linear, then, if it is known a priory that the space of solutions is rotationally invariant, then separation of variables in spherical (R,θ,φ) coordinates will work. The Laplacian is itself rotationally invariant. The equation can, of course, have other ingredients, so that the presence of the Laplacian doesn't guarantee anything by itself.

There are many other symmetries besides rotational symmetry. There is also host of orthogonal coordinate systems, some common (like (R,θ,φ)), and many less common. Naturally, the number of independent variables may be different from 3. Finding the symmetries and using the appropriate coordinate system can lead to a successful separation of variables. YohanN7 (talk) 19:54, 29 September 2012 (UTC)[reply]

Integrating Hamilton's equations to find the Hamiltonian

Ordinarily, a Hamiltonian ${\mathcal {H}}(q,p,t)$ is derived, and taking its partial derivatives yields Hamilton's equations of motion, $\partial _{p}{\mathcal {H}}={\dot {q}}$ , $-\partial _{q}{\mathcal {H}}={\dot {p}}$ . 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 $f(t)$ 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 $\partial _{p}f(q,p,t)={\dot {q}}$ and $-\partial _{q}g(q,p,t)={\dot {p}}$ , but it is not obvious if the functions $f$ and $g$ 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 $f(q,p,t)=g(q,p,t)$ 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)[reply]