Differentiation a Hamiltonian Vector Field

[Kreizhn]

Homework Statement

Consider a smooth manifold M, and smooth functions $H_i: T^*M \to \mathbb R, i=0,1$ on the cotangent bundle. Further, let $z:[t_1,t_2] \subset \mathbb R \to M$ define a trajectory on $T^*M$ by
$$\frac{dz}{dt} = \vec H_0(z(t)) + u(t) \vec H_1(z(t))$$
where $\vec H_i$ is the Hamiltonian vector field corresponding to $H_1$ That is, $\vec H_i$ is the unique vector on $T(T^*M)$ satisfying
[tex] \iota_{\vec H_i}\omega = dH_i [/itex]
where $\iota$ is the interior product and $\omega$ is the canonical symplectic form on $T^*M$.

If we know that $\left. H_1 (z(t)) = 0$ show that
$$dH_1(z(t)) \vec H_0(z(t)) + u(t) dH_1(z(t)) \vec H_1(z(t)) = 0$$

The Attempt at a Solution

Firstly, I am confused with the notation. In particular the statement
$$dH_1(z(t)) \vec H_0(z(t))$$
I think that it should be interpreted as the covector field at a fixed instance z(t) applied to the vector field at z(t); that is, in perhaps less ambiguous notation
$$(dH_1)_{z(t)}(\vec H_0(z(t)))$$
The next thing that confuses me is how to apply the chain rule to this vector field. I've tried expressing [tex] \vec H_1 [/itex] in a canonical Darboux coordinate system and then differentiating, but it doesn't seem to lead anywhere. Any input would be appreciated.

 

[mighty2000]

it is important to carefully analyze the given information and try to understand the problem before attempting to solve it. Here are some thoughts and steps that may help in approaching this problem:

1. Understand the given notation: It is important to understand the given notation and definitions before proceeding with any calculations. Take some time to review the definitions of smooth manifolds, cotangent bundles, Hamiltonian vector fields, and symplectic forms.

2. Review the properties of Hamiltonian vector fields: Remember that Hamiltonian vector fields have some special properties, such as being divergence-free with respect to the symplectic form. This may be helpful in understanding the given equation.

3. Use the given information: The given equation involves the Hamiltonian vector fields corresponding to the smooth functions H_0 and H_1. Use the given information about these functions to simplify the equation.

4. Apply the chain rule: The chain rule is a powerful tool in calculus and can be used to differentiate composite functions. In this case, we have a composite function z(t) = z(t) composed with the Hamiltonian vector field \vec H_0. Try using the chain rule to differentiate this function and see if it leads to any useful information.

5. Use the fact that H_1(z(t)) = 0: This is the key piece of information that will help us solve the problem. Use this fact to simplify the given equation and see if it leads to any useful insights.

6. Consider different coordinate systems: Sometimes, changing to a different coordinate system can make the problem easier to solve. In this case, you may want to try expressing the Hamiltonian vector fields in a different coordinate system and see if it helps in solving the problem.

7. Take your time: It is important to take your time and not rush through the problem. This will help you to carefully analyze the given information and come up with a solution. Remember, solving problems in mathematics and science requires patience and persistence.