About integrable systems

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Furthermore, since each S_i(q_i) is independent of the other coordinates, we can see that these quantities commute with each other, making them integrals of motion in involution.Therefore, we have shown that if the Hamilton Jacobi equation is separable in certain coordinates, then the system is integrable and there exist n integrals of motion in involution. This is a very useful result, as it allows us to simplify the analysis of many classical mechanical systems.
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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 [tex] n[/tex] integrals of motion in involution.


Homework Equations





The Attempt at a Solution



If the H-J equation

[tex]
H\left(q^i,\frac{\partial G}{\partial q^i}\right)=-\frac{\partial G}{\partial t}
[/tex]

is separable then it means that

[tex]
G=T(t)+Q_1(q^1)+\ldots+Q_n(q^n)
[/tex]

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

Thanks
 
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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
 

1. What is an integrable system?

An integrable system is a mathematical model that can be solved exactly using analytical methods. This means that the system can be described by a set of equations that can be solved without the need for numerical approximations. Integrable systems are often used in physics, engineering, and other fields to model complex phenomena.

2. How do integrable systems differ from non-integrable systems?

Integrable systems are characterized by having a sufficient number of conserved quantities, which are physical quantities that remain constant over time. This allows for the system to be solved using analytical methods. Non-integrable systems, on the other hand, do not have enough conserved quantities and therefore require numerical methods to be solved.

3. What are some examples of integrable systems?

One example of an integrable system is the harmonic oscillator, which describes the motion of a mass attached to a spring. Another example is the Kepler problem, which models the motion of planets around the sun. Other examples include the Toda lattice, the Korteweg-de Vries equation, and the Ising model in statistical mechanics.

4. What are the applications of integrable systems?

Integrable systems have a wide range of applications in various fields, including physics, mathematics, engineering, and biology. They are used to model physical phenomena such as fluid dynamics, quantum mechanics, and nonlinear optics. In engineering, integrable systems are used to design efficient structures and devices. They are also used in the study of biological processes, such as protein folding and gene regulation.

5. What are the challenges in studying integrable systems?

Despite their usefulness, integrable systems can be difficult to study due to their complex nature. One challenge is identifying and understanding the conserved quantities that make a system integrable. Another challenge is finding analytical solutions for these systems, which may require advanced mathematical techniques. Additionally, the behavior of integrable systems can change when perturbed or when additional interactions are introduced, making their study challenging.

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