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Time homogeneous lagrangian

  1. Oct 14, 2009 #1
    if the lagrangian is time homogenous ,the hamiltonian is a constant of the motion .
    Is this statment correct ?
  2. jcsd
  3. Oct 14, 2009 #2
    if [tex]\frac{\partial L}{\partial t}=0[/tex] then the hamiltonian is a conserved quantity. So yes. If the lagrangian doesn't explicitly depend on time, H is conserved.
  4. Oct 14, 2009 #3
    can you give me example ?
  5. Oct 14, 2009 #4
    well, the typical situation (where your coordinates are somewhat normal (ie, can be related somehow to the cartesian coordinate system in a time independent fashion) then the hamiltonian is the energy of the system.

    ie, simple harmonic oscillator:

    L=T-U= 1/2 m x'^2 - 1/2 k x^2

    where m is the mass, k is the spring constant, the first term is the kinetic energy (1/2 m v^2) and the second term is the potential (1/2 k x^2)

    in this case H=T+U = Kinetic Energy + Potential Energy = Total Energy = Constant
  6. Oct 14, 2009 #5
    Mandatory exercise: Derive Hamilton's equations and prove the result.
  7. Oct 15, 2009 #6
    how we can explain the differential of lagrangian is a perfect ?L dt
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