Explaining Time Homogeneous Lagrangian and Hamiltonian Conservation

[eman2009]

if the lagrangian is time homogenous ,the hamiltonian is a constant of the motion .
Is this statement correct ?

 

[lstellyl]

if $$\frac{\partial L}{\partial t}=0$$ then the hamiltonian is a conserved quantity. So yes. If the lagrangian doesn't explicitly depend on time, H is conserved.

 

[eman2009]

can you give me example ?

 

[lstellyl]

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

 

[Count Iblis]

Mandatory exercise: Derive Hamilton's equations and prove the result.

 

[eman2009]

how we can explain the differential of lagrangian is a perfect ?L dt