Explicitly dependent on time Lagrangian

In summary, the conversation is about the need to turn a Lagrangian into one that does not have explicit time dependence. The individual has spent a lot of time on the problem but has not found a satisfactory answer. They are wondering if there is a method or transformation that can be used to achieve this, or if it is simply a matter of trial and error. They would appreciate any help or guidance on the matter.
  • #1
Magister
83
0
I need to turn this Lagrangian in one that is not explicitly dependent on time.

[tex]
L = \frac{\alpha}{2} (q^\prime + qbe^{-\alpha t})^2-q^2 \frac{ab}{2} e^{-\alpha t} (\alpha +b e^{-\alpha t})- \frac{k q^2}{2}

[/tex]

I have already spent a lot of time around this problem but I am far from getting a satisfactory answer. Is there any method for doing this or is just by looking? Any help would be great! Thanks
 
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  • #2
Is there a transformation [tex]q ->\bar{q }= qf(\alpha + be^{-\alpha t})[/tex] that puts the explicit time dependence into q ?
 
  • #3


I understand your frustration in trying to turn an explicitly time-dependent Lagrangian into one that is not explicitly dependent on time. However, there are methods that can be used to achieve this. One approach is to use a transformation of variables, such as a canonical transformation, to eliminate the explicit time dependence in the Lagrangian. This can involve introducing new variables or changing the form of the existing ones. Another approach is to use a redefinition of the Lagrangian parameters, such as changing the constants or coefficients in the Lagrangian to remove the time dependence. Both of these methods can be complex and require careful consideration, but they can be effective in transforming the Lagrangian into a form that is not explicitly dependent on time. It may also be helpful to consult with colleagues or reference materials for additional guidance in solving this problem. Keep persevering and you will find a satisfactory solution.
 

What is an explicitly dependent on time Lagrangian?

An explicitly dependent on time Lagrangian is a mathematical function used in physics to describe the motion of a system over time. It takes into account the kinetic and potential energies of the system at any given time and allows for the prediction of the system's future behavior.

How is an explicitly dependent on time Lagrangian different from a regular Lagrangian?

An explicitly dependent on time Lagrangian takes into account the change in time and how it affects the system, whereas a regular Lagrangian does not. This allows for a more accurate prediction of the system's behavior over time, especially in systems where time plays a significant role.

What are some real-world applications of explicitly dependent on time Lagrangians?

Explicitly dependent on time Lagrangians are commonly used in physics and engineering to model the behavior of complex systems such as pendulums, satellites, and chemical reactions. They are also used in the study of celestial mechanics and quantum mechanics.

What are the advantages of using an explicitly dependent on time Lagrangian?

One of the main advantages of using an explicitly dependent on time Lagrangian is that it allows for a more accurate prediction of a system's behavior over time. It also simplifies the mathematical calculations involved in modeling complex systems and helps to identify the key factors that affect the system's behavior.

How can one derive an explicitly dependent on time Lagrangian for a specific system?

To derive an explicitly dependent on time Lagrangian for a system, one must first identify all the components of the system, their positions, and their potential and kinetic energies. Then, using the Lagrangian formalism, the time-dependent terms are added to the regular Lagrangian to account for any changes in the system over time. This results in the explicitly dependent on time Lagrangian for that particular system.

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