I Checking if a stationary point is a minimum using Lagrangian Mechanics

AI Thread Summary
To determine if a stationary point is a minimum in Lagrangian mechanics, one can use the second variation, which is analogous to the second derivative test in standard calculus. The discussion highlights a practice question where the stationary point of an integral needs to be verified as a minimum, with the initial approach involving a second derivative test on the function f(x'). The poster questions whether the condition S(a) > S_actual can be applied to I(a) in the context of Lagrangian mechanics. They express confusion about the appropriate methods for analyzing minima in this framework, indicating a need for clarification on the topic. Understanding the second variation is crucial for correctly identifying minima in Lagrangian problems.
beans123
Messages
5
Reaction score
0
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!
 
Physics news on Phys.org
Try googling "2nd variation in Lagrangian mechanics". (This is analog of 2nd derivatives in ordinary calculus.)
 
The rope is tied into the person (the load of 200 pounds) and the rope goes up from the person to a fixed pulley and back down to his hands. He hauls the rope to suspend himself in the air. What is the mechanical advantage of the system? The person will indeed only have to lift half of his body weight (roughly 100 pounds) because he now lessened the load by that same amount. This APPEARS to be a 2:1 because he can hold himself with half the force, but my question is: is that mechanical...
Let there be a person in a not yet optimally designed sled at h meters in height. Let this sled free fall but user can steer by tilting their body weight in the sled or by optimal sled shape design point it in some horizontal direction where it is wanted to go - in any horizontal direction but once picked fixed. How to calculate horizontal distance d achievable as function of height h. Thus what is f(h) = d. Put another way, imagine a helicopter rises to a height h, but then shuts off all...
Some physics textbook writer told me that Newton's first law applies only on bodies that feel no interactions at all. He said that if a body is on rest or moves in constant velocity, there is no external force acting on it. But I have heard another form of the law that says the net force acting on a body must be zero. This means there is interactions involved after all. So which one is correct?
Back
Top