Does Separability of the Hamilton-Jacobi Equation Guarantee Integrability?

[luisgml_2000]

Homework Statement

I'm asked to prove that if the Hamilton Jacobi equation is separable in certain coordinates then the system is integrable, that is, there exist $$n$$ integrals of motion in involution.

Homework Equations

The Attempt at a Solution

If the H-J equation

$$H\left(q^i,\frac{\partial G}{\partial q^i}\right)=-\frac{\partial G}{\partial t}$$

is separable then it means that

$$G=T(t)+Q_1(q^1)+\ldots+Q_n(q^n)$$

At first I thought that $$Q_i$$ would be the integrals of motion but later I realized that is not the case. What can I do next?

Thanks

 

[Jhero]

for your post! This is a very interesting topic that ties together several important concepts in classical mechanics. To prove that a system is integrable, we must show that there exist n integrals of motion in involution. In other words, we need to find n quantities that are conserved and that commute with each other.

To start, let's recall the definition of a separable Hamilton Jacobi equation. This means that the Hamiltonian can be written as a sum of a time-dependent term and n independent terms, each of which depends only on one coordinate. In other words, we can write the Hamiltonian as H(q, p) = T(t) + V_1(q_1) + ... + V_n(q_n), where T(t) is the time-dependent term and V_i(q_i) is the potential for the i-th coordinate.

Now, let's consider the total energy of the system, E = T(t) + V_1(q_1) + ... + V_n(q_n). Since the Hamiltonian is separable, we can see that E is a constant of motion, as it does not depend on any of the coordinates or momenta. This is our first integral of motion.

Next, we can use the Hamilton Jacobi equation to find additional integrals of motion. The key here is to use the fact that the Hamilton Jacobi equation is separable in certain coordinates. This means that we can write the action, S = ∫(p dq - H dt), as a sum of n independent terms, each of which depends only on one coordinate. In other words, we can write S = S_1(q_1) + ... + S_n(q_n).

Now, if we take the partial derivative of S with respect to any of the coordinates, we get: ∂S/∂q_i = p_i. This means that ∂S_i/∂q_i = p_i, where S_i(q_i) is the term in the action that depends only on q_i.

Using this result, we can rewrite the Hamilton Jacobi equation as:

H(q, ∂S_i/∂q_i) = -∂S_i/∂t

Since the Hamiltonian is separable, we can see that this equation is satisfied for each coordinate, i.e. for each S_i(q_i). This means that each S_i(q_i) is also a