# Hamilton-Jacobi theory problem

• Thales Castro
In summary, the Hamiltonian describes the trajectory of a particle on the ##xy## plane and the Hamilton-Jacobi equation can be used to find a complete integral for the system. By choosing an appropriate form for the Hamilton principal function, a solution for the equations of motion can be obtained. However, it is important to carefully consider the form of the principal function in order to avoid trivial solutions.
Thales Castro

## Homework Statement

A particle moves on the ##xy## plane having it's trajectory described by the Hamiltonian
$$H = p_{x}p_{y}cos(\omega t) + \frac{1}{2}(p_{x}^{2}+p_{y}^{2})sin(\omega t)$$

a) Find a complete integral for the Hamilton-Jacobi Equation

b) Solve for ##x(t)## and ##y(t)## with ##x(0) = y(0) = 0##; ##\dot{x}(0) = 0##, ##\dot{y}(0) = v_{0}##.

## Homework Equations

Hamilton-Jacobi equation, if ## H = H\left( \{ q_{i} \} , \{ p_{i} \} ,t \right) ##

$$H\left( \{q_{i} \} , \{ \frac{\partial S}{\partial q_{I}} \} , t \right) + \frac{\partial S}{\partial t} = 0$$

## The Attempt at a Solution

a) First, because ## H ## doesn't depend explicitly on ##x## and ##y##, the Hamilton principal function ## S ## can be written as:

$$S = \alpha_{1} x + \alpha_{2} y + \bar{S}(\alpha,t)$$

Now, all we need for ##S## to be a complete integral is

$$det\left( \frac{\partial^{2}S}{\partial q_{i} \partial {\alpha_{i}}} \right) \neq 0$$

which is immediately satisfied by our ##S##.

Now, is there any way of choosing the easiest ##\bar{S}(t)## to find the equations of motion? I tried ## \bar{S}(t) = 0 ##, but the HJ Equation got me the trivial solution ## \alpha_{1} = \alpha_{2} = 0 ##. Any ideas? Thanks in advance

Ok, I just realized that the Hamilton-Jacobi Equation was already giving me the derivative of ## \bar{S}(t) ##

## 1. What is Hamilton-Jacobi theory problem?

Hamilton-Jacobi theory problem is a mathematical concept in classical mechanics that deals with finding a solution to the Hamilton-Jacobi equation, which is a partial differential equation that describes the evolution of a physical system over time.

## 2. What is the significance of Hamilton-Jacobi theory problem?

The significance of Hamilton-Jacobi theory problem lies in its ability to provide a complete description of a physical system's dynamics, including both position and momentum, without the need for solving the equations of motion. This allows for a more efficient and elegant approach to solving problems in classical mechanics.

## 3. How is Hamilton-Jacobi theory problem related to other theories in physics?

Hamilton-Jacobi theory problem is closely related to other theories in physics, such as Hamiltonian mechanics, Lagrangian mechanics, and quantum mechanics. It can be seen as a generalization of these theories, providing a more comprehensive approach to understanding the behavior of physical systems.

## 4. What are some applications of Hamilton-Jacobi theory problem?

Hamilton-Jacobi theory problem has various applications in physics, including celestial mechanics, optics, and quantum mechanics. It is also used in engineering and economics to solve optimization problems.

## 5. What are the limitations of Hamilton-Jacobi theory problem?

One of the main limitations of Hamilton-Jacobi theory problem is that it is not applicable to systems with time-dependent potentials. It also does not take into account the effects of quantum mechanics, so it is not suitable for describing systems at the atomic or subatomic level.

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