# Lagrangian problem: Ball oscillating in spherical bowl

• Lengalicious
In summary, the conversation discusses finding the period of oscillation for a solid sphere placed in a spherical bowl using the Lagrangian approach. The equations of motion are incorrect and there are additional conditions to take into account, such as the relation between x- and y-coordinates and the relation between phi dot and x dot and y dot due to the spherical shape and rolling motion of the ball.
Lengalicious

## Homework Statement

Consider a solid sphere of radius r to be placed at the bottom of a spherical bowl radius R, after the ball is given a push it oscillates about the bottom. By using the Lagrangian approach find the period of oscillation.

## The Attempt at a Solution

Ok so this is as far as i get, not sure if what I am doing is correct:
General coords (∅,x,y)

T = 1/2*m*(xdot2+ydot2) + 1/2*I*∅dot2
T = 1/2*m*xdot2+1/2*m*ydot2+(mr22)/5 (Since Inertia of Solid Sphere is = 2/5*m*r2)
V = mgy

L = 1/2*m*xdot2+1/2*m*ydot2+(mr22)/5 - mgy

∂L/∂∅dot = 2/5*m*r2*∅dot
∂L/∂∅ = 0

∂L/∂xdot = m*xdot
∂L/∂x = 0

∂L/∂ydot = m*ydot
∂L/∂y = -mg

d(2/5*m*r2*∅dot)/dt = 0
d(m*xdot)/dt = 0
d(m*ydot)/dt = -mg

Up to here I am not sure whether I've been doing the right thing, and also, not quite sure how this helps me find the period of oscillation?

EDIT: Sigh, I know my equations of motion are wrong because i should be getting something similar to the DE for an SHO. Could really use some help.

Last edited:
There are some conditions you need to take into account: because bowl is spherical, you have a relation between x- and y-coordinates, and because the ball is rolling, there is a relation between phi dot and x dot and y dot.

Ok, but did I use the correct coordinate system? Or should it be phi and theta, and once I have the constraints how do I encorporate them into the equation of motion?

## 1. What is the Lagrangian problem and how does it relate to a ball oscillating in a spherical bowl?

The Lagrangian problem is a concept in classical mechanics that involves finding the equations of motion for a system by using the Lagrangian function. In the case of a ball oscillating in a spherical bowl, the Lagrangian function takes into account the kinetic and potential energies of the ball and the constraints imposed by the bowl's shape.

## 2. How is the Lagrangian function derived for a ball oscillating in a spherical bowl?

The Lagrangian function for a system can be derived using the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action. In the case of a ball oscillating in a spherical bowl, the action is the integral of the Lagrangian function over time.

## 3. What are the advantages of using the Lagrangian method for solving problems involving oscillating systems?

Using the Lagrangian method allows for a more systematic and efficient approach to solving problems involving oscillating systems. It takes into account all the forces and constraints acting on the system, making it easier to analyze and understand the system's behavior. It also reduces the number of variables to be considered, making the equations of motion simpler to solve.

## 4. What are some real-world applications of the Lagrangian problem with a ball oscillating in a spherical bowl?

The Lagrangian problem with a ball oscillating in a spherical bowl has many applications in physics and engineering. It can be used to analyze the motion of a pendulum, the behavior of a satellite orbiting a planet, or the motion of a molecule in a chemical reaction. It can also be applied to design and optimize oscillating systems, such as clock pendulums or suspension systems in vehicles.

## 5. Are there any limitations to using the Lagrangian method for solving problems involving oscillating systems?

The Lagrangian method is limited to systems that can be described using a small number of generalized coordinates. It also assumes that the system is conservative, meaning that energy is conserved and there are no dissipative forces. Additionally, the Lagrangian method may not be suitable for highly complex or nonlinear systems, where other methods such as numerical simulations may be more appropriate.

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