Uniform Circular Motion in Lagrangian Formalism

In summary, the problem involves finding the Lagrangian for a particle constrained to move in a circle of radius r, with relevant equations including L = T - V, where L is the Lagrangian, T is the kinetic energy, and V is the potential energy. The question asks about the potential energy associated with the centripetal force holding the particle in its circular trajectory and whether or not the equation L = T - V holds in this case. A discussion ensues about the inclusion of constraint forces in Lagrangian mechanics.
  • #1
jsc314159
21
1
Problem: Consider a particle of mass m, constrained to move in a circle of radius r. Find the Lagrangian:

Relevant Equations: L = T - V

Where L is the Lagrangian, T is the kinetic energy, and V is the potential energy.

My questions is this. T is the kinetic energy and would simply equal mV^2/2 or mr^2w^2/2 depending on the coordinate system chosen.

What about V? There has to be a force on the particle holding it in its circular trajectory or it would simply fly off. However, no central force is mentioned in the problem. For all I know the particle may be held in place by a string, or maybe it rides in a circular track. Anyhow, does it make sense to talk about a potential energy associated with centripetal force?

Is it possible that L = T - V doesn't hold in this case since the forces involved are velocity dependent (centripetal force)? I know for the EM Lagrangian L is not T - V.

jsc
 
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  • #2
It's been a while since I've done anything like this, and I don't have my copy of Goldstein handy, but doesn't Lagrange's equation also contain a term for a contraint?

That is, the full equation is:

[tex] Q = \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial q}[/tex]

Where Q is the constraint force...?
 
  • #3
Andy Resnick said:
It's been a while since I've done anything like this, and I don't have my copy of Goldstein handy, but doesn't Lagrange's equation also contain a term for a contraint?

That is, the full equation is:

[tex] Q = \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial q}[/tex]

Where Q is the constraint force...?
That's news to me. Where did you get this formula from? Forces of constraint are not explicitly given in Lagrange's equations. That's the beauty of them. It looks to me that the Q above refers to non-monogentic external forces (i.e. forces for which there is no associated potential energy function) such as friction.

Best wishes

Pete
 
  • #4
jsc314159 said:
Problem: Consider a particle of mass m, constrained to move in a circle of radius r. Find the Lagrangian:

Relevant Equations: L = T - V

Where L is the Lagrangian, T is the kinetic energy, and V is the potential energy.

My questions is this. T is the kinetic energy and would simply equal mV^2/2 or mr^2w^2/2 depending on the coordinate system chosen.

What about V? There has to be a force on the particle holding it in its circular trajectory or it would simply fly off. However, no central force is mentioned in the problem. For all I know the particle may be held in place by a string, or maybe it rides in a circular track. Anyhow, does it make sense to talk about a potential energy associated with centripetal force?

Is it possible that L = T - V doesn't hold in this case since the forces involved are velocity dependent (centripetal force)? I know for the EM Lagrangian L is not T - V.

jsc

well you're lagrangian coordinates are (\theta,r) do the transformation on velocities and you'l find T... you have a rotational symmetry so V=V(r)...
if F=mV^2/r... what is the V? remebere that F=-grad(V)...

ciao
 
  • #5
pmb_phy said:
That's news to me. Where did you get this formula from? Forces of constraint are not explicitly given in Lagrange's equations. That's the beauty of them. It looks to me that the Q above refers to non-monogentic external forces (i.e. forces for which there is no associated potential energy function) such as friction.

Best wishes

Pete

It's entirely possible I am confused. I do recall there's a way to introduce constraint forces into the Lagrange formalism (we had to do a problem from Symon- a ball rolling off another ball), but can't remember exactly what we did- my class notes are at home, buried in the basement IIRC.
 
  • #6
[tex] Q = \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}[/tex]

Sorry to be pedantic. A dot was missing.
 
  • #7
[QUOTE[tex] Q = \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}[/tex]
[/QUOTE]I still don't see where you got this from. Perhaps yo just made slight mistake. There is a similar expression in Lagrangian mechanics which is reminiscent of your equation. It differs from yours by a negative sign and the interpretation of Q. The equation I speak of is

[tex] Q_i = \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} [/tex]

where L = Lagrangian of system = T - U and Qi are the components of the generalized force. U = generalized potential aka velocity-dependant potential.

See - http://electron6.phys.utk.edu/phys594/Tools/mechanics/summary/lagrangian/lagrangian.htm

For a single charged particle moving in an electromagnetic field U = q[itex]\phi[/tex] - qv*A where q = charge of particle, v particle's velocity and A is the magnetic vector potential.

But these generalized forces are not forces of constraint. They are the total force acting on the ith particle. And yes, I was a bit wrong in my statement :redface:. That's why I love posting! I learn new things when I least expect it :smile:. They part of this force is the force of contstraint. In fact this force can be thought of as representing the sum of the external force and the .

This is found in Classical Mechanics - Third Ed., by Goldstein, Safko and Poole (2002), page 23.

In the example you gave U = V = 0.

I hope this has helped some?

Best wishes

Pete
 
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  • #8
This is found in Classical Mechanics - Third Ed., by Goldstein, Safko and Poole (2002), page 23.

In the example you gave U = V = 0.

I hope this has helped some?

Best wishes

Pete

Thanks for the replies.

I believe Pete is correct. The constraining forces are not conservative and therefore are not associated with a potential in the Lagrangian. Constraining forces can be added to the Euler-Lagrange equation as Lagrange multipliers.

jsc
 

1. What is the Lagrangian formalism?

The Lagrangian formalism is a mathematical framework used to describe the dynamics of physical systems. It is based on the principle of least action, which states that the path taken by a system between two points is the one that minimizes the action, a quantity that combines the system's kinetic and potential energy.

2. How is uniform circular motion described using Lagrangian formalism?

In uniform circular motion, an object moves in a circular path with constant speed. In Lagrangian formalism, this motion is described by considering the system's kinetic and potential energies, which are defined in terms of the position and velocity of the object. The equations of motion can then be derived by minimizing the action using the Euler-Lagrange equation.

3. What is the advantage of using Lagrangian formalism to describe uniform circular motion?

One advantage is that it allows for a more elegant and concise mathematical representation of the system's dynamics. It also provides a deeper understanding of the underlying principles governing the motion, as well as the conservation of energy and momentum. Additionally, Lagrangian formalism can be extended to more complex systems, making it a versatile tool in physics.

4. Can Lagrangian formalism be used for non-uniform circular motion?

Yes, Lagrangian formalism can be applied to any type of motion, as long as the system's kinetic and potential energies can be defined. This includes non-uniform circular motion, where the speed of the object may vary as it moves along the circular path. In such cases, the equations of motion become more complex, but the same principles still apply.

5. How does Lagrangian formalism relate to other methods of describing motion, such as Newton's laws?

Lagrangian formalism is a more general approach to describing motion compared to Newton's laws. While Newton's laws are based on the concept of forces, Lagrangian formalism is based on the principle of least action, which takes into account both kinetic and potential energies. In fact, the equations derived from Lagrangian formalism can be equivalent to those obtained from Newton's laws for simple systems. However, for more complex systems, Lagrangian formalism can be a more efficient and elegant method of analysis.

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