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Let $S$ be a Hamiltonian system with three degrees of freedom, let $q_1, q_2, q_3$ be the generalized coordinates. Suppose the Hamiltonian of the system is $$ H = \alpha \sum_{i=1}^3 p_i + V(q_1^2 + q_2^2 + q_3^2) $$ where $\alpha$ is some constant.

I need to determine all the constants of motion.

First, I noticed that since the time variable is not explicit in $H$, we have $\frac{dH}{dt} = 0$ and so energy is conserved.

Then I found the Lagrangian:

$$L = \sum_i \dot{q}_i \ p_i - H = \sum_i p_i (\dot{q}_i - \alpha) - V(q_1^2 + q_2^2 + q_3^2). $$

Not really how to proceed from this to find the constants of motion. How do I compute $$ \frac{\partial L}{\partial q_i} = ? $$

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  • $\begingroup$ I'd try changing to spherical coordinates in your Lagrangian. $\endgroup$ – Daniel Jul 22 '16 at 14:26
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Lagranges equations are: ${\partial L \over \partial q_i} = {d \over dt }{ \partial L \over \partial q_i'}$ where $q_i'={dq_i \over dt}$

You can find constants of motion using lagranges equations and Hamiltons equations. You already know that the Hamiltonian is conserved-time is not explicit. (The energy is not always equal to the Hamiltonian) You can find which other quantities are conserved using lagranges equations and Hamiltons equations. The generalized momenta are defined as:

${ \partial L \over \partial q_i'} = p_i$

You may also consider changing to spherical coordinates, then finding the angular momenta. You can do this with the transformations $q_1 \rightarrow r sin(\theta)cos(\phi), q_2 \rightarrow r cos\theta cos(\phi)$ and $q_3 \rightarrow rcos(\theta)$ (keep in mind you would have to apply the same transformations to the generalized velocities $q_1', q_2',q_3'$) From this transformation you will obtain a new Hamiltonian and lagrangian as a $f(r, r', \theta, \theta',\phi,\phi')$ with time implicit. Using noether's theorum, If a component is absent from the Lagrangian, its conjugate momenta is a conserved quantity. If you do not see $\theta, \phi$ in the lagrangian $p_\theta, p_\phi$ are conserved quantities. This is often the case with $V(r)$ where $r=q_1^2 +q_2^2+q_2^2$

In this way, you recover three new variables, and you get to check to see if the angular momenta are conserved.


Another aproach i forget to mention earlier: Use Hamiltons equations:

$-{\partial \mathcal{H} \over \partial q_i} = {dp_i \over dt}$

and

${\partial \mathcal{H} \over \partial p_i} = {dq_i \over dt}$

You can use these equations to get some more equations of motion. Then you can do some algebra and see if there are any conserved quantities.

The approach i would take would be to 1. Apply transformation to spherical coordinates then 2. look for conserved angular momenta


I edited this post in response to the mod's guidance.

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  • $\begingroup$ Now it should be more along the line of what OP wanted $\endgroup$ – Haru Fujimura Jul 22 '16 at 17:34
  • $\begingroup$ Thanks for the reply. But how do I find $p_i$ in terms of the spherical polar coordinates though? I still have $p_i$ in my Lagrangian, when it should depend only on generalized velocities and coordinates. $\endgroup$ – Kamil Jul 22 '16 at 22:52
  • $\begingroup$ So you are not given, $p_i$ then? $\endgroup$ – Haru Fujimura Jul 23 '16 at 0:13
  • $\begingroup$ OP, can you give us any more information about the problem? Is this a single particle system? Or is this just a math problem Hamiltonian, that doesnt necessarily correspond to a physical system? $\endgroup$ – Haru Fujimura Jul 23 '16 at 0:37
  • $\begingroup$ Where is this problem from? Maybe that will help us solve it. $\endgroup$ – Haru Fujimura Jul 23 '16 at 0:43

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