Rolling Object on Curved Surface: Lagrangian Mechanics + Constraint

In summary, the conversation is about plotting the trajectory of a ball rolling without slip on a curved surface. The known variables include the radius, mass, and moment of inertia of the ball, as well as formulas for the curvature of the path and the relationship between path length and height above the ground. The homework equations involve the Euler-Lagrange Equations. The attempt at a solution has led to an answer for the acceleration of the ball along the path length, but it is not a function of the starting height as expected. It is suspected that an incorrect assumption was made, and further discussion leads to the realization that the assumption about the linear decrease in height with arc length is incorrect. The potential term is expected to be a complicated function of
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Homework Statement



I want to be able to plot a trajectory wrt time of a ball that rolls without slip on a curved surface.

Known variables:
-radius/mass/moment of inertia of the ball.
-formula for the curvature of the path (quadratic)
-formula relating path length and corresponding height above the ground. (linearly decreasing, i.e. as you travel down the curve, you decrease in height).

Homework Equations


Euler-Lagrange Equations

The Attempt at a Solution


See attached for further details.

I've come up with an answer for the acceleration of the ball along the path length. However, it isn't a function of the starting height as I expected. I suspect I may have made an assumption that I'm not supposed to but I can't see what it is.
 

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After some though and comparison to classical mechanics, I've determined that the assumption that the height of the ball decreases linearly with arc length is not correct.

It seems my potential term is going to be some complicated function of arc length.

Anyone else have thoughts?
 

1. What is the concept of Lagrangian mechanics?

Lagrangian mechanics is a mathematical approach used to study the motion of a system of particles. It is based on the principle of least action, which states that a system will follow the path that minimizes the total energy or action. This approach is commonly used in physics and engineering to analyze complex systems.

2. How does Lagrangian mechanics apply to rolling objects on curved surfaces?

In the context of rolling objects on curved surfaces, Lagrangian mechanics is used to determine the equations of motion for the object. The rolling object is treated as a system of particles, and the Lagrangian is defined as the difference between the kinetic and potential energies of the system. This approach takes into account the constraints imposed by the curved surface and allows for a more accurate analysis of the motion of the object.

3. What is a constraint in the context of Lagrangian mechanics?

A constraint is a condition or limitation that restricts the motion of a system. In the case of rolling objects on curved surfaces, the constraint is the shape of the surface. This means that the object must follow a specific path determined by the curvature of the surface. Constraints are taken into account when defining the Lagrangian and solving for the equations of motion.

4. How does the Lagrangian equation account for constraints?

The Lagrangian equation includes terms that take into account the constraints of the system. These terms, known as Lagrange multipliers, are added to the equation to ensure that the constraints are satisfied. By using these multipliers, the equations of motion can be solved accurately, taking into account the constraints imposed by the curved surface.

5. What are some applications of Lagrangian mechanics in real-world scenarios?

Lagrangian mechanics has many applications in various fields, including physics, engineering, and robotics. It is commonly used to analyze the motion and stability of systems such as pendulums, satellites, and robots. It is also used in the study of celestial mechanics, where it is used to model the motion of planets and other celestial bodies. In addition, Lagrangian mechanics has applications in the development of control systems for vehicles and robots.

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