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| '''Generating functions''' which arise in [[Hamiltonian mechanics]] are quite different from [[generating functions]] in mathematics. In physics, a generating function acts as a bridge between two sets of canonical variables when performing a [[canonical transformation]].
| | Let me initial begin by introducing myself. My title is Boyd Butts although it is not the title on my beginning certificate. Years in over the counter std test ([http://www.gaysphere.net/blog/262619 have a peek here]) past he moved to North Dakota and his family loves it. Supervising is my profession. To gather coins is what his family and him enjoy. |
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| ==Details==
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| There are four basic generating functions, summarized by the following table:
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| {| border="1" cellpadding="5" cellspacing="0"
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| ! style="background:#ffdead;" | Generating Function
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| ! style="background:#ffdead;" | Its Derivatives
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| |-
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| |<math>F= F_1(q, Q, t) \,\!</math>
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| |<math>p = ~~\frac{\partial F_1}{\partial q} \,\!</math> and <math>P = - \frac{\partial F_1}{\partial Q} \,\!</math>
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| |-
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| |<math>F= F_2(q, P, t) - QP \,\!</math>
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| |<math>p = ~~\frac{\partial F_2}{\partial q} \,\!</math> and <math>Q = ~~\frac{\partial F_2}{\partial P} \,\!</math>
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| |-
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| |<math>F= F_3(p, Q, t) + qp \,\!</math>
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| |<math>q = - \frac{\partial F_3}{\partial p} \,\!</math> and <math> P = - \frac{\partial F_3}{\partial Q} \,\!</math>
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| |-
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| |<math>F= F_4(p, P, t) + qp - QP \,\!</math>
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| |<math>q = - \frac{\partial F_4}{\partial p} \,\!</math> and <math> Q = ~~\frac{\partial F_4}{\partial P} \,\!</math>
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| |}
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| ==Example==
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| Sometimes a given Hamiltonian can be turned into one that looks like the [[harmonic oscillator]] Hamiltonian, which is
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| :<math>H = aP^2 + bQ^2.</math>
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| For example, with the Hamiltonian
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| :<math>H = \frac{1}{2q^2} + \frac{p^2 q^4}{2},</math>
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| where ''p'' is the generalized momentum and ''q'' is the generalized coordinate, a good canonical transformation to choose would be
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| {{NumBlk|:|<math>P = pq^2 \text{ and }Q = \frac{-1}{q}. \,</math>|{{EquationRef|1}}}}
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| This turns the Hamiltonian into
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| :<math>H = \frac{Q^2}{2} + \frac{P^2}{2},</math>
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| which is in the form of the harmonic oscillator Hamiltonian.
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| The generating function ''F'' for this transformation is of the third kind,
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| :<math>F = F_3(p,Q).</math>
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| To find ''F'' explicitly, use the equation for its derivative from the table above,
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| :<math>P = - \frac{\partial F_3}{\partial Q},</math>
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| and substitute the expression for ''P'' from equation ({{EquationNote|1}}), expressed in terms of ''p'' and ''Q'':
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| : <math>\frac{p}{Q^2} = - \frac{\partial F_3}{\partial Q}</math>
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| Integrating this with respect to ''Q'' results in an equation for the generating function of the transformation given by equation ({{EquationNote|1}}):
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| |<math>F_3(p,Q) = \frac{p}{Q}</math>
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| |}
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| To confirm that this is the correct generating function, verify that it matches ({{EquationNote|1}}):
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| : <math>q = - \frac{\partial F_3}{\partial p} = \frac{-1}{Q}</math>
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| ==See also==
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| *[[Hamilton-Jacobi equation]]
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| *[[Poisson bracket]]
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| ==References==
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| *{{cite book | author=Goldstein, Herbert | title=Classical Mechanics | publisher=Addison Wesley | year=2002 | isbn=978-0-201-65702-9}}
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| [[Category:Classical mechanics]]
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| [[Category:Hamiltonian mechanics]]
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| {{classicalmechanics-stub}}
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Let me initial begin by introducing myself. My title is Boyd Butts although it is not the title on my beginning certificate. Years in over the counter std test (have a peek here) past he moved to North Dakota and his family loves it. Supervising is my profession. To gather coins is what his family and him enjoy.