# Solving the equations of motions of a particles constrained to a sphere.

• VoxCaelum
In summary, the problem involves a particle moving on a two dimensional sphere of radius r, with a Lagrangian given by L(q,q')=\frac{1}{2}m\sum(qi')2. The first part of the problem involves transforming the Lagrangian into spherical coordinates and writing down the resulting Euler-Lagrange equations. The second part involves solving these equations and writing down the solution in the original coordinates, including the integration constants denoted as the initial position and velocity. The equations involve some typos and constraints due to the motion being constrained to the surface of a sphere, but can be solved by considering the derivative of the Lagrangian with respect to one of the variables as an integral of motion.
VoxCaelum

## Homework Statement

Consider a particle moving on a two dimensional sphere of radius r, whose Lagrangian is given by:
L(q,q')=$\frac{1}{2}$m$\sum$(qi')2

(A) Transform the Lagrangian into spherical coordinates and write down the resulting Euler-Lagrange equations.

(B) Solve the Euler-Lagrange equations corresping to the Lagrangian and write down the solution in the original coordinates (qi). Express your answer to include the integration constants and denote them as the initial position and velocity.

## Homework Equations

$\frac{∂L}{∂q}$=$\frac{d}{dt}$$\frac{∂L}{∂q'}$

## The Attempt at a Solution

I think I have part (A) solved. The problem is part B. I really don't know where to start. But this is what I have for part (A)

mr2$\phi$''Sin2(θ)+$\phi$'θ'2Sin(θ)Cos(θ)=0
r2$\phi$'Sin(θ)Cos(θ)=mr2θ''

I would not know where to start solving these equations.

First of all, you have some typos in your equations, since 2nd equation has 1 time derivative in left and 2 in right side. Also m should drop out everywhere.

Secondly, you forgot to calculate the last equation (wrt r), which will probably prove to be very useful in solving the set.

In the second equation I forgot a power of $\phi$'.
Also I did not forget to do one of the E.L. equations. Because we are constrained to the surface of a sphere I simply set r'=0 and treated r as a constant parameter instead of a dynamical variable. This reduces the Lagrangian (in spherical coordinates) to a Lagrangian involving two functions instead of three.
L=$\frac{1}{2}$m(r2θ'2+r2$\phi$'2Sin2(θ))

I also failed to notice that since the derivative of the lagrangian with respect to $\phi$ is zero one of the E.L. equations will yield an integral of motion which should help out a bit.

## What is the purpose of solving equations of motion for particles constrained to a sphere?

The purpose of solving equations of motion for particles constrained to a sphere is to understand the motion and behavior of particles that are confined to a spherical surface. This can be useful in various fields of science, such as physics, chemistry, and biology, where particles may be constrained to move on or within a spherical structure.

## What are the equations of motion for particles constrained to a sphere?

The equations of motion for particles constrained to a sphere involve the use of spherical coordinates, which include the radial distance, polar angle, and azimuthal angle. These coordinates are used to describe the position, velocity, and acceleration of the particle as it moves on the surface of the sphere. The equations of motion also take into account any forces acting on the particle, such as gravity or normal forces.

## What are some common methods for solving equations of motion for particles constrained to a sphere?

Some common methods for solving equations of motion for particles constrained to a sphere include using Lagrange multipliers, using Newton's laws of motion, and using vector calculus techniques. These methods can help to simplify the equations and make them easier to solve, depending on the specific constraints and forces acting on the particle.

## What are some applications of solving equations of motion for particles constrained to a sphere?

Solving equations of motion for particles constrained to a sphere has many practical applications. For example, it can be used to model the motion of planets and satellites orbiting a spherical object, or to study the motion of microscopic particles within a spherical cell or organism. It can also be used in engineering and robotics to design and control spherical robots or vehicles.

## What are some challenges in solving equations of motion for particles constrained to a sphere?

One of the main challenges in solving equations of motion for particles constrained to a sphere is accounting for all the forces and constraints acting on the particle. This may require advanced mathematical techniques and can become more complex as the number of particles or constraints increases. Additionally, accurately measuring and accounting for factors such as friction and air resistance can also pose challenges in solving these equations.

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