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  • 7/24/2019 Liouvilles Theorem

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    LIOUVILLES THEOREM

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    LIOUVILLES THEOREM

    The dynamical state of a system at some instant of

    be represented by a point in the phase space.

    The point ill not be stationary b!t ill mo"e alon#

    tra$ectory hich is determined from the e%!ations o

    &here '(' )%i**.%f+ pi **..pf, is the 'amiltoni

    system.

    As a res!lt of motion+ the density of the system in

    space chan#es ith time.

    At a #i"en point in phase space+ for hich !se is ma

    -io!"illes theorem.

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    The theorem consists of two parts:

    )/,The first part states the conser"ation of denphase space.

    )0,The second part the conser"ation in phase sp

    ) (1+

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    First part

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    No of points enterin# thro!#h AB (

    No of points enterin# thro!#h AD (

    No of points lea"in#s thro!#h 3D (.dp4

    No of points lea"in#s thro!#h B3 (5 d% 4 5=p ==========

    AB 53D (5 dp 55 dp.5 dpd%=q q .q=.=.=.=.=.=.=.=.=.=.Remainin# points in AB3D (6 5 5 AD5B3 (5 d% 55 d%.5 dp d%=p p .p=.=.=.=.=.=.=.=.=.=.

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    Remainin# points in AB3D(6 4

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    We know that Hamiltonian equation

    q

    (p

    Take positial difference w.r.t q

    7rom

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    Ta8e positial difference . r.t p

    7rom

    S!bstit!te in

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    This form of e9pression may be called the principle of the

    of principle inphase space.

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    )0, Second part: 7or this part+ e to pro"e that

    V, (1

    &e 8no that (5===========

    Since the n!mber of phase pointsThe phase space m!st remain fi9ed the system can neither

    created nor destroyed; e therefore ha"e

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    &e ha"e pro"ed that

    'a"e it follos that

    5

    Since 5

  • 7/24/2019 Liouvilles Theorem

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    7olloin#

  • 7/24/2019 Liouvilles Theorem

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    THANKING YOU


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