Lagrangian remains invariant under addition

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SUMMARY

The Lagrangian remains invariant under the addition of an arbitrary function of time due to the nature of the Euler-Lagrange equations, which only involve derivatives with respect to position and velocity. This means that the total time derivative of any function added to the Lagrangian does not affect the equations of motion. This principle is foundational for advanced concepts such as contact transformations and the Hamilton-Jacobi theory, which simplify classical physics problems by allowing the addition of specific functions whose derivatives meet certain criteria.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with Euler-Lagrange equations
  • Knowledge of classical mechanics principles
  • Basic concepts of Hamilton-Jacobi theory
NEXT STEPS
  • Study the derivation and implications of the Euler-Lagrange equations
  • Explore contact transformations in classical mechanics
  • Learn about Hamilton-Jacobi theory and its applications
  • Investigate the role of total time derivatives in Lagrangian formulations
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Physicists, students of classical mechanics, and researchers interested in advanced theoretical physics concepts such as Lagrangian invariance and Hamiltonian dynamics.

preet0283
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what is the reason that the lagrangian remains invariant under addition of an arbtrary function of time?
 
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Hi,
It is the equations of motion that are invariant under the addition of a function that is the total time derivative of some function, to the Lagrangian. Since the Euler-Lagrange equations involve derivatives with respect to position and velocity only, a partial derivative wrt to position or velocity of this added function will be zero.
Hope this helps

Ray
 
Preet, what you just noticed is the basis for later things like contact transformations and the resulting Hamilton-Jacobi theory. It turns out that you can add a more general class of functions whose derivatives obey a certain relationship, and if you can find these functions and changes of variable then you can make any classical physics problem a piece of cake.
 

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