Checking if a stationary point is a minimum using Lagrangian Mechanics

In summary, the conversation discusses how to determine if a stationary point is a minimum using Euler's equation and second derivative tests. It also mentions the concept of 2nd variation in Lagrangian mechanics as an alternative method of solving the problem. The person is new to Lagrangian mechanics and is seeking clarification and advice on the topic.
  • #1
beans123
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I'm having trouble understanding how to find out whether or not a stationary point is a minimum and I'm hoping for some clarification. In my class, we were shown that, using Euler's equation, the straight-line path:
Screenshot 2023-02-05 18.16.34.png

with constants a and b results in a stationary point of the integral:
Screenshot 2023-02-05 18.16.47.png

A certain practice question then asks to show that the stationary point corresponds to a minimum. My only attempt so far was performing a simple second derivative test on the function f(x') which turned out to be successful. However, I'm wondering if this is the only way to solve such a problem. I know that a minimum is satisfied if S(a) > S_actual, but can that same idea be mapped onto I(a), that is, is a minimum achieved if I(a) > I_actual (if that even makes sense)? I'm very new to Lagrangian mechanics and find it kind of overwhelming so forgive me if this is a silly question. It just seems that I took the calculus way of solving this when that may not be the ideal method for a class based on Lagrangian mechanics/. I appreciate any help/advice!
 
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  • #2
Try googling "2nd variation in Lagrangian mechanics". (This is analog of 2nd derivatives in ordinary calculus.)
 
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1. What is a stationary point in Lagrangian Mechanics?

In Lagrangian Mechanics, a stationary point refers to a point in the system where the first derivative of the Lagrangian function with respect to the generalized coordinates is equal to zero. This indicates that the system is in equilibrium and the potential and kinetic energies are balanced.

2. How do you check if a stationary point is a minimum using Lagrangian Mechanics?

To check if a stationary point is a minimum using Lagrangian Mechanics, you need to calculate the second derivative of the Lagrangian function with respect to the generalized coordinates. If the second derivative is positive, then the stationary point is a minimum. If it is negative, then the stationary point is a maximum. If the second derivative is zero, then further analysis is needed.

3. What is the significance of finding a minimum in Lagrangian Mechanics?

Finding a minimum in Lagrangian Mechanics indicates that the system is in a state of stable equilibrium. This means that the system will tend to return to this state if it is disturbed. It also means that the potential energy of the system is at its lowest point, which is desirable for many physical systems.

4. Can Lagrangian Mechanics be used to find stationary points in all physical systems?

Yes, Lagrangian Mechanics can be used to find stationary points in all physical systems. However, it is most commonly used in systems with multiple degrees of freedom and in systems with constraints, such as those found in classical mechanics and electromagnetism.

5. Are there any limitations to using Lagrangian Mechanics to find stationary points?

One limitation of using Lagrangian Mechanics to find stationary points is that it assumes the system is conservative, meaning that there is no external force acting on the system. Additionally, it may be more complex and time-consuming to use Lagrangian Mechanics in systems with many degrees of freedom, and numerical methods may be needed to solve the resulting equations.

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