Transforming Lagrangian without changing the equations of motion.

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SUMMARY

The discussion centers on the transformations of the Lagrangian in classical mechanics, specifically focusing on canonical transformations and their impact on the equations of motion. It is established that adding a total time derivative and multiplying the Lagrangian by a constant are valid transformations that do not alter the solutions of the equations of motion. Additionally, the modified Hamilton's principle defines canonical transformations as those that satisfy the equation pq - H = PQ - K + (total time derivative). The inquiry also seeks to explore further transformations of the Lagrangian without changing variables.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with Hamilton's principle
  • Knowledge of canonical transformations
  • Basic grasp of variational principles in physics
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  • Study the implications of canonical transformations in Lagrangian mechanics
  • Explore the modified Hamilton's principle in depth
  • Research additional transformations of the Lagrangian beyond total time derivatives
  • Learn about the role of symmetries in classical mechanics
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This discussion is beneficial for physicists, particularly those specializing in classical mechanics, as well as students and researchers interested in advanced topics related to Lagrangian and Hamiltonian formulations.

alemsalem
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I know that it works with adding a total time derivative and multiplying the Lagrangian by a constant.
are these the only things that can be done to a Lagrangian such that the equations of motion have the same solutions q(t).

Thanks!
 
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There is a wider class of so called canonical transformations.
 
Dickfore said:
There is a wider class of so called canonical transformations.

These are the ones I'm having a problem with. the modified Hamilton's principle gives this definition for a canonical transformation:
pq - H = PQ - K + (total time derivative).. and that's because you can add a total time derivative inside the integral for the action.
I also want to know if there is more transformations on the Lagrangian without a change of variables.

Thanks :)
 

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