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Lagrangian problem

  1. Mar 30, 2012 #1
    1. The problem statement, all variables and given/known data
    A Lagrangian for a particular physical system can be written as,

    [itex]L^{\prime }=\frac{m}{2}(a\dot{x}^{2}+2b\dot{x}\dot{y}+c\dot{y}^{2})-\frac{K%
    }{2}(ax^{2}+2bxy+cy^{2})[/itex]

    where a and b are arbitrary constants but subject to the condition that b2
    -ac≠0.What are the equations of motion?Examine particularly two cases a=0=c and b=0,c=a.What is the physical system described by above lagrangian.? What is the significance for the condition b2-ac?

    2. The attempt at a solution

    I've done the mathematics.But donno the physics!

    Equations of motion are,
    [itex]ma\ddot{x}+mb\dot{y}+Kax+Kby=0[/itex]

    [itex]ma\ddot{y}+mb\dot{x}+Kcy+Kbx=0[/itex]

    I think these equations represent coupled 2D harmonic oscillator.(i'm not sure)

    when a=c=0,
    [itex]mb\dot{y}+Kby=0[/itex]
    [itex]mb\dot{x}+Kbx=0[/itex]

    when b=0,c=-a,

    [itex]ma\ddot{x}+Kax=0[/itex]
    [itex]ma\ddot{y}+Kay=0[/itex]

    Thanks.
     
  2. jcsd
  3. Mar 30, 2012 #2
    Hello! In general, the system described by the lagrangian represents two "independent" harmonic oscillators, in a sense I'm going to explain: you can collect the coordinates in a vector [itex]\vec{v}=(x,y)^t[/itex] and the coefficient a,b,c in a matrix M such that [itex]M_{11}=a[/itex], [itex]M_{12}=M_{21}=b[/itex], [itex]M_{22}=c[/itex]; the lagrangian takes the following form:

    [itex]L=\frac{m}{2}\dot{\vec{v}^t}M\dot{\vec{v}}-\frac{K}{2}\vec{v}^t M \vec{v}[/itex].

    Since M is symmetric, we diagonalize it through an orthogonal matrix O:
    [itex]M=O^t M^{\text{diag}} O[/itex].

    We can now define two new coordinates x' and y' that can be incorporated in a vector [itex]\vec{v'}=(x',y')^t[/itex] which is equal by definition to

    [itex]\vec{v'}=O\vec{v}[/itex].

    In this case the lagrangian has manifestly the form of two decoupled harmonc oscillators (if b^2-ac different from zero).
    The significance of b^2-ac different from zero means that the two eigenvalues of M are different from zero and, so there are two modes which oscillate.

    As far as I know (and if I don't forget any hypothesis), this is a quite general feature of lagrangian which are at most quadratic in the coordinates. I hope this is right and the answer you need,
    Francesco
     
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