- #1
stunner5000pt
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The Lagrangian of a aprticle of charge q and mass m is given by
[tex] L = \frac{1}{2} m \dot{r}^2 - q \phi(r,t) + \frac{q}{c} r \bullet A(r,t) [/tex]
a) determine the Hamiltonain function H(r,p) (in terms of r and its conjugate momentum p) and explain waht conditions the electric and mangetic field intensities
[tex] E = - \frac{\partial \phi}{\partial r} - \frac{1}{c} \frac{\partial A}{\partial t} [/tex]
[tex] B = \frac{\partial}{\partial r} \times A [/tex]
must satisfy if the Hamiltonain is to be a constant of motion
I am not sure if the Hamiltonian is
[tex] H = \dot{r} \frac{\partial L}{\partial \dot{r}} -L [/tex]
OR [tex] H = r \frac{\partial L}{\partial r} -L [/tex]
[tex] \dot{r} \frac{\partial L}{\partial \dot{r}} = m \dot{r}^2 + \frac{q}{c} \dot{r} \bullet A(r,t) [/tex]
[tex] H = \dot{r} \frac{\partial L}{\partial \dot{r}} -L = m \dot{r}^2 + \frac{q}{c} \dot{r} \bullet A(r,t) - \frac{1}{2} m \dot{r}^2 - q \phi(r,t) + \frac{q}{c} r \bullet A(r,t) [/tex]
[tex] H = \frac{1}{2} m\dot{r}^2 + q \phi(r,t) [/tex]
[tex] H(r,p) = \frac{1}{2} p \dot{r} + q \phi(r,t) [/tex]
for H to be a constant of motion, then dh/dt = 0 ,yes?
[tex] \frac{dH}{dt} = \frac{1}{2} \dot{p} \dot{r} + \frac{1}{2} p \ddot{r} + q \frac{\partial \phi}{\partial t} [/tex] (not sure about the last term)
[tex] \frac{dH}{dt} = F \frac{\dot{r}}{2} + p \frac{\ddot{r}}{2} + q \frac{\partial \phi}{\partial t} [/tex]
im not sure how the E and B field come into play because none of their terms appear in the equation for dH/dt
b) Work out Hamilton's equations of motion for this system (i.e. the first order equations of r(t),p(t)), and show taht htey are equivalent to the second order Euler Lagrange equation for this system.
I am currently on this...
Your help is greatly appreciated
[tex] L = \frac{1}{2} m \dot{r}^2 - q \phi(r,t) + \frac{q}{c} r \bullet A(r,t) [/tex]
a) determine the Hamiltonain function H(r,p) (in terms of r and its conjugate momentum p) and explain waht conditions the electric and mangetic field intensities
[tex] E = - \frac{\partial \phi}{\partial r} - \frac{1}{c} \frac{\partial A}{\partial t} [/tex]
[tex] B = \frac{\partial}{\partial r} \times A [/tex]
must satisfy if the Hamiltonain is to be a constant of motion
I am not sure if the Hamiltonian is
[tex] H = \dot{r} \frac{\partial L}{\partial \dot{r}} -L [/tex]
OR [tex] H = r \frac{\partial L}{\partial r} -L [/tex]
[tex] \dot{r} \frac{\partial L}{\partial \dot{r}} = m \dot{r}^2 + \frac{q}{c} \dot{r} \bullet A(r,t) [/tex]
[tex] H = \dot{r} \frac{\partial L}{\partial \dot{r}} -L = m \dot{r}^2 + \frac{q}{c} \dot{r} \bullet A(r,t) - \frac{1}{2} m \dot{r}^2 - q \phi(r,t) + \frac{q}{c} r \bullet A(r,t) [/tex]
[tex] H = \frac{1}{2} m\dot{r}^2 + q \phi(r,t) [/tex]
[tex] H(r,p) = \frac{1}{2} p \dot{r} + q \phi(r,t) [/tex]
for H to be a constant of motion, then dh/dt = 0 ,yes?
[tex] \frac{dH}{dt} = \frac{1}{2} \dot{p} \dot{r} + \frac{1}{2} p \ddot{r} + q \frac{\partial \phi}{\partial t} [/tex] (not sure about the last term)
[tex] \frac{dH}{dt} = F \frac{\dot{r}}{2} + p \frac{\ddot{r}}{2} + q \frac{\partial \phi}{\partial t} [/tex]
im not sure how the E and B field come into play because none of their terms appear in the equation for dH/dt
b) Work out Hamilton's equations of motion for this system (i.e. the first order equations of r(t),p(t)), and show taht htey are equivalent to the second order Euler Lagrange equation for this system.
I am currently on this...
Your help is greatly appreciated
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