Lagrangian mechanics of a bead of mass

[stunner5000pt]

consider a bead of mass m constrained to move on a fricitonless wire helix whose equations in cylindrical polar coords is
z = a phi where a is some constant
the bead is acted upon by a force which deends on the distance from the cneter only.
Formulate the problem using s the distance along the helix as your generalized coordinate.

for distance s along the helix $s = a \phi_{0}$
but r would not be constnat
$$T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi_{0}}^2 + \dot{z}^2)$$
z dot is zero and phi0 is a constnat soso
$$T = \frac{1}{2} m \dot{r}^2$$


the force is dependant on the distance from the center only
thus $$\vec{F} = -k\vec{r} = -\nabla V$$(say)
then $$V = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + a^2 \phi_{0}^2)$$
$$L = T - V = \frac{1}{2} m\dot{r}^2- \frac{1}{2} k (r^2 + a^2 \phi_{0}^2)$$
is this formulation correct?
Or am i totally off?

 

[dextercioby]

First question...Why doesn't "s" denote the generalized coordinate in the lagrangian...?

L should be $L\left(s,\dot{s}\right)$...

Daniel.

 

[Meir Achuz]

consider a bead of mass m constrained to move on a fricitonless wire helix whose equations in cylindrical polar coords is
z = a phi where a is some constant
the bead is acted upon by a force which deends on the distance from the cneter only.
Formulate the problem using s the distance along the helix as your generalized coordinate.

for distance s along the helix $s = a \phi_{0}$
but r would not be constnat
$$T = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi_{0}}^2 + \dot{z}^2)$$
z dot is zero and phi0 is a constnat soso
$$T = \frac{1}{2} m \dot{r}^2$$
the force is dependant on the distance from the center only
thus $$\vec{F} = -k\vec{r} = -\nabla V$$(say)
then $$V = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + z^2) = \frac{1}{2} k (r^2 + a^2 \phi_{0}^2)$$
$$L = T - V = \frac{1}{2} m\dot{r}^2- \frac{1}{2} k (r^2 + a^2 \phi_{0}^2)$$
is this formulation correct?
Or am i totally off?

You are totally off.
1. For a helix in cylindrical coords, z and phi vary but r is constant.
2. I don't know what "a force which depends on the distance from the center only" means. Does "center" mean the axis of the helix or does it mean the origin of coordinates?
(I assume gravity is not acting.)
3. The statement "a force which depends on the distance from the center only" does not mean that F=-kr. It just means that F depends only the magnitude of the distance from whatever "center" means.
4. After you have written down the Lagrangian, eliminate phi and z in terms of s.

 

[stunner5000pt]

well our prof told us to assume that F = -kr maybe i shouldve stated taht anyway
s is the disatnce along the helix...
would this have something to do with arc length?
something along the lines of this
$$s = \int_{t_{1}}^{t_{2}} \sqrt{(\frac{\partial f}{\partial r})^2 + (\frac{\partial f}{\partial r})^2 + (\frac{\partial f}{\partial r})^2}$$
where f is the function of the length of hte arc...?

 

[Meir Achuz]

If s is the length along the helix, it probably is
$$s=\phi\sqrt{a^2+z^2}=$$,
with [tex]z=sa/\sqrt{r^2+a^2}[\tex]
and [tex]\phi=s/\sqrt{r^2+a^2}[\tex]
The r in F=-kr must be the distance from one point on the axis.
(Is it?)
Otherwise the force could never do anything.