(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A charged particle is constrained to move in a plane under the influence of a central force potential V=1/2kr^2 and a constand magnetic field B perpendicular to the plane, so that [tex]A=\frac{1}{2}\vec B \times \vec{\dot{r}}[/tex]. We want to solve the problem with the Hamilton-Jacobi method, but my problem focuses elsewhere.

2. Relevant equations

The lagrangian is

[tex]L=\frac{1}{2}m\dot{r}^2+q\vec A \cdot \vec{\dot{r}} -\frac{1}{2}kr^2[/tex]

I'm interested in the second term. If [tex]\vec B=B \hat z[/tex], then the term is written

[tex]\frac{qB}{2}(-y\dot{x}+x\dot{y})[/tex]

or, using polar coordinates,

[tex]\frac{qB}{2}r^2\dot{\theta}[/tex]

Now, the conjugate momentum is

[tex]p_{\theta}=\frac{\partial L}{\partial\theta}=...+\frac{qB}{2}r^2[/tex]

If one tries to find the Hamiltonian, such as

[tex]H=p_r\dot{r}+p_{\theta}\dot{\theta}-L[/tex]

finds that this term disappears, which is at least strange.

So, am I missing something fundamental here? Thanx in advance!

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# Homework Help: Goldstein 3rd ed, 10.6

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