Hamilton-jacobi and action angle

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We have$$E=\frac{m}{2} (v_0^2+\omega^2 q_0^2), \quad Q=\frac{1}{\omega} \arcsin \left (\frac{q_0}{\sqrt{q_0^2+(v_0/\omega)^2}} \right ).$$So in summary, the Hamilton-Jacobi equation is a way to find new canonical coordinates that describe the motion of a system. It is equivalent to demanding that the new Hamiltonian vanishes, and for systems with symmetry, this can lead to a simplified solution for the equations of motion.
  • #1

i am a MS physics student. We are following the goldstein book for classical mech and have exams next week.

Does anyone know of any good (easy - "how to do it" procedure explained) notes or pdf hand outs on the Hamilton Jacobi eq and the action angle variables.

I do not have time to study too deeply - I only want to get the hang of it and know how to do the sums.

If you know any such problem solving oriented notes pls post the links here or in my inbox.

Alternatively you can write a coupel of lines here yourself explaining how to go about the sums -

EG - for lagrangian -

1 - find the degrees of freedoms -
2 - for each one choose a generalized coordinate/variable.
3 - find out the PE and the KE
4 - The Lagrangian then is L = KE -PE
5 - solve the EL eq for each generalized coordinate.

Something very simplistic like this - Thanks a lot.
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  • #2
I don't know, what you mean by "sums". The concept of the Hamilton-Jacobi Partial Differential Equation (HJPDE) is to look for new canonical coordinates ##(Q_i,P_i)## such that the motion is described by ##(Q_i,P_i)=\text{const}.## (let me warn you that in almost all cases there is no solution, but only for "integrable systems", i.e., systems where you have sufficient symmetry so that you have enough conserved quantities according to Noether's theorem). One can show that this is equivalent to the demand that the new Hamiltonian vanishes:

To that end you look for the corresponding generator of the canonical transformation. We choose the generator in the form ##g(q,P,t)##. Then
$$p=\partial_q g, \quad H'(Q,P,t)=H(q,p,t)+\partial_t g \stackrel{!}{=}0.$$
This leads to the HJPDE,
$$H \left (q,\frac{\partial g}{\partial p},t \right ) + \frac{\partial g}{\partial t}=0.$$
If ##H## is not explicitly time dependent, we know that the energy is conserved. Then
$$\frac{\partial g}{\partial t}=-H=-E \; \Rightarrow \; g(P,q,t)=-E t + S(P,q).$$
The HJPDE then simplifies to
$$H \left (q,\frac{\partial g}{\partial q} \right )=E.$$
Let's look at the harmonic oscillator as the most simple example. The Hamiltonian is
$$H=\frac{p^2}{2m}+\frac{m \omega^2}{2} q^2.$$
We can apply the simplified version, because energy is conserved since ##H## doesn't depend explicitly on time. Thus ##E## is the new canonical momentum, and we have the HJPDE
$$\frac{1}{2m} \left (\frac{\partial S}{\partial q} \right )^2 + \frac{m \omega^2}{2} q^2=E.$$
We can easily find a solution, because it's just an integral:
$$\frac{\partial S}{\partial q}=\sqrt{2 m E-m^2 \omega^2 q^2}$$
with the solution
$$S(q,E)=\int \mathrm{d} q \sqrt{2 m E-m^2 \omega^2 q^2}.$$
We don't need to consider an ##E##-dependent integration constant, because this only adds another ##E##-dependent constant to the constant ##Q##, which we get from
$$g=-E t+S(q,E) \; \Rightarrow \; Q=\frac{\partial g}{\partial E}=-t+\frac{1}{\omega} \arcsin \left (\sqrt{\frac{m \omega^2}{2E}} q \right ).$$
Resolved to ##q##:
$$q=\sqrt{\frac{2E}{m \omega^2}} \sin[\omega(t+Q)].$$
Of course, ##E## and ##Q## are determined by the initial conditions ##q(t_0)=q_0## and ##v(t_0)=\dot{q}(t_0)=v_0##.

1. What is Hamilton-Jacobi theory?

Hamilton-Jacobi theory is a mathematical framework used to describe the dynamics of a physical system. It is based on the principle of least action, which states that a system will follow the path that minimizes the action (a quantity related to energy) of the system. This theory is commonly used in classical mechanics and has applications in quantum mechanics as well.

2. What is the Hamiltonian in Hamilton-Jacobi theory?

The Hamiltonian is a function that represents the total energy of a system in Hamilton-Jacobi theory. It is a sum of the kinetic and potential energies of the system and is used to describe the evolution of the system over time.

3. How does Hamilton-Jacobi theory relate to action-angle variables?

In Hamilton-Jacobi theory, the action-angle variables are a set of canonical coordinates that can be used to describe the motion of a system. These variables have the property that the action is constant along the trajectory of the system, while the angle changes linearly with time.

4. What is the significance of the Hamilton-Jacobi equation?

The Hamilton-Jacobi equation is a partial differential equation that is used to solve for the action function in Hamilton-Jacobi theory. It is significant because it allows for the transformation of a system's equations of motion into a simpler form, making it easier to find solutions.

5. What are some applications of Hamilton-Jacobi theory?

Hamilton-Jacobi theory has many applications in physics, engineering, and other fields. It is commonly used in celestial mechanics to study the motion of planets and other celestial bodies. It also has applications in quantum mechanics, where it is used to solve the Schrödinger equation for certain systems. In addition, Hamilton-Jacobi theory is used in control theory and optimal control problems in engineering.

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