Explaining Time Homogeneous Lagrangian and Hamiltonian Conservation

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SUMMARY

The discussion confirms that if the Lagrangian is time homogeneous, the Hamiltonian is indeed a conserved quantity. Specifically, when the partial derivative of the Lagrangian with respect to time is zero (\(\frac{\partial L}{\partial t}=0\)), the Hamiltonian remains constant throughout the motion. An example provided is the simple harmonic oscillator, where the Lagrangian \(L = T - U = \frac{1}{2} m x'^2 - \frac{1}{2} k x^2\) leads to a Hamiltonian \(H = T + U\) that represents the total energy of the system, which is conserved. The discussion also emphasizes the importance of deriving Hamilton's equations to validate these results.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with Hamiltonian mechanics
  • Knowledge of simple harmonic motion
  • Basic calculus, particularly differentiation
NEXT STEPS
  • Study the derivation of Hamilton's equations in detail
  • Explore the implications of time homogeneity in various physical systems
  • Investigate the relationship between Lagrangian and Hamiltonian formulations
  • Examine examples of conserved quantities in different mechanical systems
USEFUL FOR

Physicists, mechanical engineers, and students of classical mechanics who are interested in the principles of conservation laws in Lagrangian and Hamiltonian frameworks.

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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