Mass on a string.lagrange mechanics

In summary, the mass is free to move on a string that is rotating at angular velocity w. The only degrees of freedom are the position r and the momentum pr. The kinetic energy L is given by:
  • #1
tmoan
39
0
hi there i am studying lagrange and hamiltonian mechanics.
i came about this question from a previous test session please if anyone can help.

the question is about a mass rotating on a rectilinear string attached without friction from a point A on the z axis.
the length of the string is h = OA (O being the origin of the 3D system)

the mass is free to move on the string as a pearl on a necklace for example.

the rotational vel. "w" is given constant which makes the angle ф on the xy plane = wt
this means that ф is not a degree of freedom

i do not undersand however why the angle α between the string and the z axis is not.
in the solution r the position vector from A to the mass M is the only degree of freedom.

i think the answer would be simple and that if i scratched my head a bit i would know but i am still trying to figure it out in the mean time i tried asking here so..
if anybody can help me out it would be great thanks.
 
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  • #2
So, the string is rotating at angular velocity w and you have a mass M on the string that can freely move along the string?

Then you can just write down the Lagrangian which only consists of a kinetic energy term:

L = 1/2 M [r-dot^2 + r^2 w^2]

Then write down the Euler-Lagrange equatons:

d/dt dL/dr-dot - dL/dr = 0

But it is easier to construct the Hamiltonian and then use that it is conserved. The canonical momentum corresponding to r is:

pr = dL/dr-dot = M r-dot (no surprise here)

The Hamiltonian is by definition:

H = pr r-dot - L = 1/2 M r-dot^2 - 1/2 M r^2 w^2

The total derivative of H w.r.t. t is zero as easily follows in general from Hamilton's equation (provided L does not depend explicitely on time). So, you can equate H to a consant and integrate the first order diff. equation.
 
  • #3
yes thanks for the reply.
i understand what you are sayying.
but my question is why isn't the angle alpha a degree of freedom which will require its own lagrange equation right.
secondly, 1/2M r-dot^2 is for the kinetic energy of a rotating body where r is the position from the origin while here r is from pt A to M (M is where the COM of the mass is) shouldn't i do some kind of mathematical transition to A.i didn go past that yet,
i know that a hamiltonian eq would be more in handy but i am answering a question here that requires a lagrange equation first,

i think there must be a relation between angle phi and angle alpha but i don't seem to find it out yet,
i know that i only need r using my common sense but i am required to state why alpha and phi are not degrees of freedom.
i hope i made myself clear and sorry if this area is not for course work i didn't realize that it is not when i posted.
thanks for your help
 
  • #4
I completely ignored the part about the angle alpha.
If the string were to make some angle alpha with the z-axis while it rotates, then the kinetic energy is:

1/2 M [r-dot^2 + r^2 sin^2(alpha) w^2 + r^2 alpha-dot^2]

I think this is quite obvious.

So, you then write down the Euler-Lagrange equations for this system. If alpha is not assumed to be constant, then you have to deal with the equation:

d/dt [dL/d alpha-dot] - dL/d alpha = 0
 
  • #5
point taken and true.
but the fact that w is constant makes wt =phi implies phi is no longer a degree of freedom.
while phi is not constant!
in the same context why isn't alpha a degree of freedom it was never given that it is constant
but alpha must change w.r.t. phi in some way which itself changes w.r.t. w i.e. a constant and that I think is the best answer. or the PhD holder who wrote this doesn't deserve his degree for not emphasizing enough i have solved 3 other questions that are much more tiresome than this and still didn't exactly figure out why it is not clear enough.

thanks again for the help
 
  • #6
Sometimes questions are not formulated in a clear way. You should simply focus on trying to master the theory and skip questions if they are unclear. Or just replace the unclearly formulated problem with a clearly defined problem and solve that.
 

FAQ: Mass on a string.lagrange mechanics

What is the Lagrange mechanics method?

The Lagrange mechanics method is a mathematical approach used to describe the dynamics of a system of particles or rigid bodies. It is based on the principle of least action and is used to determine the equations of motion for a system.

How is the Lagrange mechanics method different from Newton's laws of motion?

The Lagrange mechanics method is a more general and elegant approach compared to Newton's laws of motion. It uses a single equation, the Lagrange equation, to describe the dynamics of a system, while Newton's laws use three separate equations. The Lagrange method also takes into account constraints and allows for the use of generalized coordinates instead of Cartesian coordinates.

What is the significance of the Lagrangian in Lagrange mechanics?

The Lagrangian is a function that represents the difference between the kinetic and potential energies of a system. It is used to derive the Lagrange equation, which is then used to determine the equations of motion for a system. The Lagrangian is also useful in solving problems involving conservation of energy and momentum.

Can the Lagrange mechanics method be applied to systems with multiple particles?

Yes, the Lagrange mechanics method can be applied to systems with multiple particles. In such cases, the Lagrangian is the sum of the individual Lagrangians for each particle, and the Lagrange equation becomes a set of equations, one for each particle.

What is the significance of the Lagrange point in celestial mechanics?

The Lagrange point, also known as a libration point, is a special point in the orbital plane of two celestial bodies where the gravitational forces of the two bodies cancel out. Objects placed at these points would remain in a stable position relative to the two bodies. The Lagrange points have been used for satellite placement and space exploration missions.

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