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T-7
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Hello folks,
I could really do with a few hints with this. (As soon as possible!)
For a repeated line of cells, two 'equations of motion' can be written:
C_{n}\dot{U_{n+1}} = I_{n}-I_{n+1}
L_{n}\dot{I_{n}} = U_{n} - U_{n+1}
where C_{n} is the capacitance, U_{n+1} the voltage after the nth cell, I_{n}-I_{n+1} the charging current, L_{n} the inductance.
Work out the Lagrangian that generated these equations.
You should find that
E = \sum_{n}\left( (1/2)\L_{n}\dot{Q_{n}}^{2} + (1/2)C_{n}U_{n+1}^{2} \right)
and
H = (1/2)\sum_{n}\left( \frac{P_{n}^{2}}{L_n}} + \frac{(Q_{n+1}-Q_{n})^{2}}{C_{n}} \right)
Note that
Q_{n} = -\int I_{n} dt => C_{n}U_{n+1} = Q_{n+1} - Q_{n}
I have suggested a Lagrangian of
L = (1/2)L_{n}\dot{Q_{n}}^{2} + Q_{n}(U_{n+1}-U_{n}) + (1/2)C_{n}\dot{U_{n+1}^{2}} + U_{n+1}(I_{n}-I_{n+1})
(which can be turned into a sum for all the repeated circuits)
but it doesn't seem convincing, despite the fact that you can recover the original 'equations of motion' using Euler-Lagrange equations (using Q and U).
I'm not sure how the energy is being derived from the Lag. in this case (in mechanics, it was always the case that L = T - U, hence E = T + U), and it seems clear that you aren't going to get either that expression for the energy or that Hamiltonian from my guess-work Lagrangian.
Any suggestions?
Many thanks!
I could really do with a few hints with this. (As soon as possible!)
Homework Statement
For a repeated line of cells, two 'equations of motion' can be written:
C_{n}\dot{U_{n+1}} = I_{n}-I_{n+1}
L_{n}\dot{I_{n}} = U_{n} - U_{n+1}
where C_{n} is the capacitance, U_{n+1} the voltage after the nth cell, I_{n}-I_{n+1} the charging current, L_{n} the inductance.
Work out the Lagrangian that generated these equations.
You should find that
E = \sum_{n}\left( (1/2)\L_{n}\dot{Q_{n}}^{2} + (1/2)C_{n}U_{n+1}^{2} \right)
and
H = (1/2)\sum_{n}\left( \frac{P_{n}^{2}}{L_n}} + \frac{(Q_{n+1}-Q_{n})^{2}}{C_{n}} \right)
Homework Equations
Note that
Q_{n} = -\int I_{n} dt => C_{n}U_{n+1} = Q_{n+1} - Q_{n}
The Attempt at a Solution
I have suggested a Lagrangian of
L = (1/2)L_{n}\dot{Q_{n}}^{2} + Q_{n}(U_{n+1}-U_{n}) + (1/2)C_{n}\dot{U_{n+1}^{2}} + U_{n+1}(I_{n}-I_{n+1})
(which can be turned into a sum for all the repeated circuits)
but it doesn't seem convincing, despite the fact that you can recover the original 'equations of motion' using Euler-Lagrange equations (using Q and U).
I'm not sure how the energy is being derived from the Lag. in this case (in mechanics, it was always the case that L = T - U, hence E = T + U), and it seems clear that you aren't going to get either that expression for the energy or that Hamiltonian from my guess-work Lagrangian.
Any suggestions?
Many thanks!