Lagrangian at a rolling sphere

In summary, the conversation discusses the momentum of a sphere rolling on an inclined wall at an angle θ. The equation for the momentum, represented by L, is given as 1/2 mv^2 + 1/5 mv^2 + mgx sin(θ) = 7/10 m v^2 + mgx sin(θ). The partial derivative of L with respect to v is denoted as p, and is equal to 7/5 mv. However, the speaker is confused as to why the factor of mv is 7/5 and not the expected value of 1. The other speaker clarifies that p is not the traditional linear momentum, but the
  • #1
sgh1324
12
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A sphere is rolling inclined wall (θ radian).
and the momentum of that sphere is

L = 1/2 mv^2 + 1/5 mv^2 + mgx sin(θ) = 7/10 m v^2 + mgx sin(θ)
∂L/∂v = p
7/5 m v = p

but I can't understand why the factor of mv is 7/5.

p is the linear momentum of sphere.

which means the factor of mv must be always 1 (my thought)

p = mv

↑that was an unchanging truth of my physical world.

why rolling effect to the momentum? angular momentum of the sphere has no relation with p direction!
 

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  • #2
sgh1324 said:
∂L/∂v = p
7/5 m v = p

but I can't understand why the factor of mv is 7/5.

p is the linear momentum of sphere.
Well, p isn't really the momentum from Newtonian analysis. It is the conjugate momentum from Lagrangian analysis. They have similar names, but they are only equal if ##KE=0.5 m v^2##
 
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1. What is Lagrangian at a rolling sphere?

Lagrangian at a rolling sphere is a mathematical concept used to describe the motion of a sphere that is rolling without slipping. It takes into account both the kinetic energy and potential energy of the sphere, as well as any external forces acting on it.

2. How is Lagrangian at a rolling sphere calculated?

Lagrangian at a rolling sphere is calculated using the Lagrangian equation, which takes into account the kinetic and potential energy of the rolling sphere. It also considers any external forces or constraints acting on the sphere. The resulting equation is then used to derive the equations of motion for the rolling sphere.

3. What is the significance of Lagrangian at a rolling sphere?

Lagrangian at a rolling sphere is an important concept in physics as it allows us to accurately describe the motion of a rolling sphere. It takes into account all relevant factors, such as energy and external forces, and can be used to predict the future motion of the sphere.

4. Can Lagrangian at a rolling sphere be applied to real-world situations?

Yes, Lagrangian at a rolling sphere can be applied to real-world situations. It is commonly used in the study of mechanics and is particularly useful in analyzing the motion of objects such as balls, wheels, and rolling cylinders.

5. What are some limitations of using Lagrangian at a rolling sphere?

While Lagrangian at a rolling sphere is a powerful tool for analyzing the motion of a rolling sphere, it does have some limitations. It assumes that the sphere is a perfect, rigid object and does not take into account factors such as friction or air resistance. Additionally, it may become more complex to use when dealing with more complex systems or objects with irregular shapes.

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