1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lagrangian condition 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%

    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,


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

    when a=c=0,

    when b=0,c=-a,


  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


    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,
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Lagrangian condition problem
  1. Lagrangian Problem (Replies: 1)

  2. Lagrangian problem (Replies: 3)

  3. Problem on Lagrangian (Replies: 5)

  4. Lagrangian Problem (Replies: 7)