Interpretation of the Lagrangian

In summary, the conversation discusses a variational problem with an integrand that does not depend on the independent variable x. This leads to an Euler equation that can be integrated to a constant, which can be interpreted as the energy of the system when x is considered as time and the integrand is the Lagrangian. This constant represents a symmetry of the action.
  • #1
Logarythmic
281
0
I have showed that if the integrand [tex]f[/tex] in the variational problem
[tex]\delta (\int f dx ) = 0[/tex]
does not depend explicitly on the independent variable x, i.e. satisfies [tex]f = f(y, \dot{y})[/tex], then the Euler equation can be integrated to

[tex]\dot{y} \frac{\partial f}{\partial \dot{y}} - f = const.[/tex]

How can I give an interpretation of this constant for the case that [tex] f = L[/tex] is the Lagrangian and [tex]x = t[/tex] is the time?
 
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  • #3
So I can interpret this as that the Energy of the system is constant?
 
  • #4
Yes, of course. It's a symmetry of the action.

Daniel.
 

Related to Interpretation of the Lagrangian

1. What is the Lagrangian?

The Lagrangian is a mathematical function that is used to describe the dynamics of a physical system. It is commonly denoted by the symbol L and is defined as the difference between the kinetic and potential energy of a system.

2. How is the Lagrangian used in physics?

The Lagrangian is used in physics to describe the motion of a system in terms of its generalized coordinates and velocities. By using the Lagrangian, we can derive the equations of motion for a system and solve for the behavior of the system over time.

3. What is the significance of the Lagrangian?

The Lagrangian is significant because it allows us to analyze complex systems and derive their equations of motion without having to use the more traditional approach of Newton's laws of motion. It also allows for a more elegant and concise representation of the dynamics of a system.

4. What is the difference between the Lagrangian and the Hamiltonian?

The Lagrangian and the Hamiltonian are both mathematical functions used to describe the dynamics of a system. However, the Lagrangian is based on the concept of energy, while the Hamiltonian is based on the concept of the system's total energy. The Hamiltonian also takes into account the effects of external forces, while the Lagrangian does not.

5. Can the Lagrangian be used in all physical systems?

Yes, the Lagrangian can be used in all physical systems as long as the system can be described using generalized coordinates and velocities. However, the Lagrangian approach may not always be the most efficient or appropriate method for certain systems, and other methods may be used instead.

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