Lagrangian equations - ring which is sliding along a wire

In summary: If not, is there a reason you expect that motion, or is there another motion that you could consider?I expect the ring to move along the wire in a spiral.In summary, the homework statement asks for help with solving an equation for a ring sliding on a wire in a gravity field. The equation is solved with Lagrangian equations, and the student is wondering if they need the Lagrangian. They also ask for help with understanding what ##\partial\Phi_\alpha\over \partial x_j## means, and for clarification on how many constraints are present.
  • #1
Oomph!
55
0

Homework Statement


Hello. I have this problem:
I have a ring which is sliding along a wire in the shape of a spiral because of gravity.
Spiral (helix) is given as the intersection of two surfaces: x = a*cos(kz), y = a*sin(kz). The gravity field has the z axis direction.
I have to find motion equations and find the wire reaction as a function of time.

2. Homework Equations

I have to solve it with this equation:
equation.png


The Attempt at a Solution


This is the first time, when I solve a example with Lagrangian equations, so I am not sure what to do.
I created this equations:
solution.jpg


I know, it is not the end. I have to find λ, motion equations and the wire reaction. But firstly, please, tell me, if I am right or where is mistake and why. Then I will continue.

Thank you very much.
 

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  • #2
Hi,
can you tell me what
your Lagrangian is ?
your generalized coordinates are ?
all variables and given/known data are, i.e. what your relevant equation symbols represent ?
Then we can continue. :wink:
 
  • #3
Do I really need the Lagrangian? When I read the study text, there was Lagrangian only in Lagrange's equations of the second kind, not in Lagrange's equations of the first kind. And I wrote there Lagrange's equations of the first kind, because in my task was, that I have to use this.

My study text is in different language, but the type of equations is same like this on 3rd page: https://www.physast.uga.edu/ag/uploads/2017%20SPRING%20-%20PHYS8011%20-%20HMWK%2004%20-%20Lagrange%20Eqs%20of%201st%20Kind.pdf

Sorry, I am confused, it is new for me.
 
  • #4
Well, I'm grateful because I had to find out about these first kind Lagrange equations (look suspiciously like Newton's) . At least that revealed what ##\Phi## stands for: the constraint equations. And they are supposed to come in the form ##\Phi_\alpha = 0##. Can you make that explicit for me, so we can check your ##\partial\Phi_\alpha\over \partial x_j## ?

Furthermore: I see only one ##\lambda##. How many constraints do you have ?

Another tack (problem solving skills): approaching this from the other end: what kind of motion do you expect ? Do your intermediate equations fit that ?
 
Last edited:

What is the Lagrangian equation for a ring sliding along a wire?

The Lagrangian equation for a ring sliding along a wire is given by L = T - V, where T is the kinetic energy of the ring and V is its potential energy.

How is the kinetic energy of the ring calculated in the Lagrangian equation?

The kinetic energy of the ring is calculated as 1/2 * m * v^2, where m is the mass of the ring and v is its velocity.

What is the potential energy term in the Lagrangian equation?

The potential energy term in the Lagrangian equation is the product of the ring's mass, gravitational acceleration, and its height above the ground.

How is the Lagrangian equation used to describe the motion of the ring?

The Lagrangian equation is used to derive the equations of motion for the ring by setting up the Euler-Lagrange equation and solving for the ring's position and velocity as functions of time.

What is the significance of the Lagrangian equation in physics?

The Lagrangian equation is a fundamental tool in classical mechanics that allows for a more efficient and elegant description of the motion of objects. It also helps to simplify complex systems and allows for the conservation of energy and momentum to be easily analyzed.

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