#### Problem

For the "falling chain" problem described in class, show that the Lagrangian is hyperregular. Compute the canonical one form $\Theta$, the canonical symplectic form $\Omega$, and the Liouville volume form $\Lambda$ on $T^*Q$. Compute the Lagrangian one form $\Theta_L$ and the Lagrangian two form $\Omega_L$ on $TQ$. Compute the action $A$, the energy $E$, and (revisiting an earlier homework problem) the Hamiltonian $H$ associated with the Lagrangian $L$. Compute the Lagrangian vector field $Z_L$ on $TQ$ and the Hamiltonian vector field $X_H$ on $T^*Q$.

#### Solution

The Lagrangian $L$ for this problem was determined by Andy for HW1:

ยต = mass per unit length

l = length of the chain

x = displacement of the chain from the x-axis

g = gravity

In order to show that the $L$ is hyperregular, one needs to check that the fiber derivative, $\mathbb F L$, is a diffeomorphism ($\mathbb F L$ is bijective and $\mathbb F L$ and $\mathbb F L^{-1}$ are both smooth).

First make sure that $\mathbb {F} L$ is locally invertible:

(2)Clearly, this is non-zero for all x not equal to zero. For this problem, $\mathbb F L$ and $\mathbb F L^{-1}$ are found to be:

(3)Clearly, $\mathbb F L$ is bijective. However, $\mathbb F L^{-1}$ is not smooth at $x = 0$. Therefore, I feel like $L$ is not hyperregular for this problem.

Moving on, the canonical one form $\Theta$, the canonical symplectic form $\Omega$, and the Liouville volume form $\Lambda$ on $T^*Q$ are computed as follows:

(5)The Lagrangian one form $\Theta_L$ and the Lagrangian two form $\Omega_L$ on $TQ$ are:

(8)Next, the action $A$, the energy $E$, and the Hamiltonian $H$ associated with the Lagrangian $L$are:

(10)As expected, the Hamiltonian found using this method matches what was found by Andy in HW1-2.

Now for the Hamiltonian vector field $X_H$ on $T^*Q$:

(13)Now for the Lagrangian vector field $Z_L$ on $TQ$:

(15)Therefore, by inspection $Z_L$ is found to be:

(17)#### Discussion

I am not sure if my conclusion that $L$ is not hyperregular is correct because usually when a problem says show something, if you show that it is not something, you screwed up…

Also, I'm not 100% sure about the Liouville volume form, but I think that it is just equal to $\Omega$ since n = 1 for this problem.

Lastly, I'm a little confused about $Z_L$ and $X_H$. While looking up Hamiltonian Vector field, I found the relation that I used to solve for it $\left(X_H = \left( \frac{\partial H}{\partial p_x}, - \frac{\partial H}{\partial x} \right)\right)$. I assumed that $Z_L$ would be similar, but w/ $L, x, \dot x$ replacing $H, x, p_x$. However, I could not find confirmation of this, so I worked it out the way we did it in class. After finishing the solution, I noticed that if I did not pull the $4 \mu x$ out, $Z_L$ would look like:

(18)which is what I expected based on the method I found/used for $X_H$. Anyone have any thoughts on this?