Lagrangean and non inertial frame

In summary: So in reality you are doing two different integrals, one in the radial direction and one in the tangential direction. Consequently, the answer you get is the result of two different integrals, and not the sum of the two integrals.
  • #1
LCSphysicist
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Homework Statement
Smooth rod OA of length l rotates around a point O in a horizontal plane with a
constant angular velocity $\delta$. The bead is fixed on the rod at a distance a from
the point O. The bead is released, and after a while, it is slipping off the rod. Find its
velocity at the moment when the bead is slipping?
Relevant Equations
.
I have tried to solve this problem using the lagrangean approach: $$L = T - V = m((\dot r)^2 + (r \dot \theta)^2)/2 - 0 = m((\dot r)^2 + (r \delta)^2)/2 - 0 $$

The problem is that the answer i got is the right answer at the smooth rod referencial, that is, at the non inertial frame.

Now we can easily translate my answer of v to the inertial frame, but that is not the point of my question. The point is, when did i have assumed i was at the rod frame? That is, i have just written the kinect energy in polar coordinates, so instead of $$\sum m x_i x^{i} / 2$$ i have written it in polar language. All of sudden, am i now in a non inertial frame?

Do polar coordinates are by definition non inertial?

Should have a potential here? I can't see where does it come from. In fact, another point i can't understand is, in a inertial frame there is not a potential energy, and in fact i have used (V=0). But yet, my answer is given in a non inertial frame...
 
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  • #2
Herculi said:
i have written it in polar language
I don't speak Inuit, but fortunately a Lagrangian is valid in any language. And in any coordinate system: inertial, non-inertial, constrained, whatever. As long as you use 'something' correctly to express kinetic energy and potential energy, Euler-Lagrange equations hold.

##\ ##
 
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  • #3
BvU said:
I don't speak Inuit, but fortunately a Lagrangian is valid in any language. And in any coordinate system: inertial, non-inertial, constrained, whatever. As long as you use 'something' correctly to express kinetic energy and potential energy, Euler-Lagrange equations hold.

##\ ##
Oh yes, i know it works in any reference frame. In fact, as i said in my question and probably was not clear, i got the right answer using my approach (but my answer was right with respect to the rod frame, that is, the rotating frame). My question is: When did i have assumed that i was at the rod frame?. In another words: I have just written the lagrangian i thought was right, got the answer, but the answer i got indicates i have used the legrangian in a non inertial frame. Even so i don't know where did i have assumed that i was at the non inertial frame itself, since i have just write the lagrangian in polar coordinates.
 
  • #4
There is no non-inertial frame at work here. What you have done is to compute ##\dot r##, the radial velocity of the bead, but the bead also has a component in the tangential direction that you cannot ignore.
 
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What is a Lagrangean frame?

A Lagrangean frame is a type of reference frame used in physics, specifically in the study of mechanics. It is a coordinate system in which the equations of motion can be described using the Lagrangian formalism, which is a mathematical framework for analyzing the dynamics of a system based on its potential and kinetic energies.

What is a non-inertial frame?

A non-inertial frame is a reference frame that is accelerating or rotating with respect to an inertial frame. In other words, the laws of motion observed in a non-inertial frame are not the same as those observed in an inertial frame, making it more complicated to analyze the dynamics of a system in a non-inertial frame.

What is the difference between a Lagrangean and non-inertial frame?

The main difference between a Lagrangean frame and a non-inertial frame is that the equations of motion in a Lagrangean frame can be described using the Lagrangian formalism, while the equations of motion in a non-inertial frame are more complex and require additional terms to account for the non-inertial effects.

Why do we use Lagrangean frames and non-inertial frames?

Lagrangean frames and non-inertial frames are used in physics to study the dynamics of systems that are not in a state of rest or uniform motion. These frames allow us to analyze the behavior of objects in non-inertial situations, such as rotating or accelerating reference frames, and can provide more accurate results than using inertial frames alone.

What are some real-world applications of Lagrangean and non-inertial frames?

One of the most well-known applications of Lagrangean and non-inertial frames is in the study of celestial mechanics, where these frames are used to analyze the motion of planets, moons, and other celestial bodies. They are also used in the design and analysis of vehicles, such as airplanes and spacecraft, which experience non-inertial effects during flight. In addition, Lagrangean and non-inertial frames are used in the study of fluid mechanics, such as in the analysis of ocean currents and weather patterns.

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