Hamilton-Jacobi theory problem

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.
  • #1
Thales Castro
11
0

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
 
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  • #2
Ok, I just realized that the Hamilton-Jacobi Equation was already giving me the derivative of ## \bar{S}(t) ## :rolleyes:
 

Related to Hamilton-Jacobi theory problem

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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