Explaining Time Homogeneous Lagrangian and Hamiltonian Conservation

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Discussion Overview

The discussion centers on the relationship between time homogeneity in the Lagrangian formulation of mechanics and the conservation of the Hamiltonian. Participants explore the implications of a time-independent Lagrangian and provide examples to illustrate the concept.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Mathematical reasoning

Main Points Raised

  • One participant asserts that if the Lagrangian is time homogeneous, then the Hamiltonian is a constant of motion.
  • Another participant clarifies that if the partial derivative of the Lagrangian with respect to time is zero, then the Hamiltonian is conserved, supporting the initial claim.
  • A request for an example is made to illustrate the concept further.
  • A participant provides an example involving the simple harmonic oscillator, explaining that in this case, the Hamiltonian represents the total energy of the system, which remains constant.
  • One post suggests a mandatory exercise to derive Hamilton's equations and prove the conservation result.
  • A question is raised about explaining the differential of the Lagrangian in relation to a specific expression involving time.

Areas of Agreement / Disagreement

Participants generally agree on the relationship between time homogeneity of the Lagrangian and the conservation of the Hamiltonian, though the discussion includes requests for examples and further clarification, indicating some areas of exploration and uncertainty.

Contextual Notes

Some assumptions regarding the conditions under which the Hamiltonian is conserved may not be fully explored, and the discussion does not resolve the mathematical steps involved in deriving Hamilton's equations.

Who May Find This Useful

Readers interested in classical mechanics, particularly those studying Lagrangian and Hamiltonian formulations, may find this discussion relevant.

eman2009
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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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