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!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook