Explaining Time Homogeneous Lagrangian and Hamiltonian Conservation

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If the Lagrangian is time-homogeneous, the Hamiltonian is indeed a conserved quantity, as indicated by the condition ∂L/∂t = 0. This means that in systems where the Lagrangian does not explicitly depend on time, the Hamiltonian represents a constant of motion. A common example is the simple harmonic oscillator, where the Hamiltonian corresponds to the total energy of the system, combining kinetic and potential energy. The discussion also touches on deriving Hamilton's equations to further validate this relationship. Understanding these principles is crucial for analyzing dynamic systems in classical mechanics.
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if the lagrangian is time homogenous ,the hamiltonian is a constant of the motion .
Is this statement correct ?
 
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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.
 
can you give me example ?
 
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
 
Mandatory exercise: Derive Hamilton's equations and prove the result.
 
how we can explain the differential of lagrangian is a perfect ?L dt
 
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