How to get the Lagrangian that generates certain diff equation ?

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In summary, the conversation discusses the process of finding a Lagrangian that generates a specific differential equation. The problem is known as the 'inverse problem of variational calculus' and can be solved by using references such as N. Akhiezer's book "The Calculus of Variations" and R. Santilli's books "Foundations of Theoretical Mechanics II" and "Foundations of Theoretical Mechanics I". These references provide examples and methods for finding the Lagrangian, including Douglas approach and inverse Legendre transformation.
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how to get the Lagrangian that generates certain diff equation ?

I have an ordinary differential equation of second order. I am looking for the Lagrangian(s) for which this equation is the Euler-Lagrange equation. I need a practical reference specifically written for one degree of freedom (one equation), I'm not interested in general theorems of existence or highly geometrical language. I found some reference by Darboux in 1894 but its in French. Any english reference?
 
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Got it. The problem is known as 'inverse problem of variational calculus'. For future references:

N. Akhiezer, "The Calculus of Variations", 1962:

page 166: example of 1D problem

R. Santilli, "Foundations of Theoretical Mechanics II", Springer-Verlag, 1983:

page 353: reworking the same example as Akhiezer
page 315: by knowing one first integral, Kobussen's method

R. Santilli, "Foundations of Theoretical Mechanics I", Springer-Verlag, 1978:

page 201: Douglas approach for n-dimension, easily applied to n=1
page 208: first obtaining Hamiltonian and then get the Lagrangian through inverse Legendre transformation
 
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smallphi said:
Got it. The problem is known as 'inverse problem of variational calculus'. For future references:

N. Akhiezer, "The Calculus of Variations", 1962:

page 166: example of 1D problem

R. Santilli, "Foundations of Theoretical Mechanics II", Springer-Verlag, 1983:

page 353: reworking the same example as Akhiezer
page 315: by knowing one first integral, Kobussen's method

R. Santilli, "Foundations of Theoretical Mechanics I", Springer-Verlag, 1978:

page 201: Douglas approach for n-dimension, easily applied to n=1
page 208: first obtaining Hamiltonian and then get the Lagrangian through inverse Legendre transformation

Thanks for the pointers!
 

1. How do you determine the Lagrangian that generates a specific differential equation?

The Lagrangian is determined by setting up the Euler-Lagrange equations, which are derived from the principle of least action. This involves finding the partial derivatives of the Lagrangian with respect to the variables in the differential equation and setting them equal to zero. Solving these equations will give the Lagrangian function.

2. Can any differential equation be generated by a Lagrangian?

No, not all differential equations can be generated by a Lagrangian. The differential equation must have certain properties, such as being second order and having a unique solution, in order for a Lagrangian to be able to generate it.

3. Are there any techniques that can help in finding the Lagrangian for a given differential equation?

Yes, there are a few techniques that can aid in finding the Lagrangian. One method is to use symmetries and conservation laws to determine the form of the Lagrangian. Another approach is to use the inverse problem method, where the Lagrangian is determined by solving a boundary value problem.

4. Is the Lagrangian unique for a given differential equation?

No, the Lagrangian is not unique for a given differential equation. There can be multiple Lagrangians that can generate the same differential equation, but they may differ in terms of their functional form or constants.

5. Can the Lagrangian be used to solve the differential equation?

Yes, the Lagrangian can be used to solve the differential equation, but it is not always the most efficient method. In some cases, the Lagrangian approach can simplify the problem and lead to a more elegant solution, but in other cases, it may be more complicated than other methods such as direct integration or numerical techniques.

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