The Lagrangian Solution of an LC Circuit

In summary, the conversation discusses the use of the Lagrangian formulation of mechanics to solve a simple LC circuit with 1 inductor and 1 capacitor. The inductor energy is treated as kinetic energy and the capacitor energy as potential energy. There is some confusion about whether the inductor energy should also be considered potential energy, but it is ultimately determined that it doesn't matter what term is used. When there are three or more terms, signs must be taken into consideration. It is also noted that capacitors can store energy permanently while inductors can only store it temporarily.
  • #1
nmbr28albert
13
3
One way to solve the simple LC circuit with 1 inductor and 1 capacitor is to use the Lagrangian formulation of mechanics and consider charge [itex] q [/itex] as the generalized coordinate. When writing down your Lagrangian, the energy of the inductor [itex] \frac{1}{2}L(\frac{dq}{dt})^2 [/itex] is treated as the kinetic energy, and the energy of the capacitor [itex] \frac{q^2}{2C} [/itex] is treated as the potential. My first instinct is to treat the inductor energy as also potential energy. Why is it considered kinetic?
 
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  • #2
Because it is similar to the kinetic energy i.e the term (dq/dt)^2 is similar to v^2. The definition of velocity v in lagrangian mechanics is the first derivative wrt time, of the generalized coordinate right?
 
  • #3
I don't think it matters which you call what.

You have two terms. The sum is the Hamiltonian (total energy) and the difference is the Lagrangian. The sign of the difference shouldn't make any difference.

When you have three or more terms, you need to be more careful with signs.
 
  • #4
that sounds right

on linear motion along coordinate x, velocity is dx/dt, i.e., the first derivattive. Kinetic energy is (1/2)mv^2 or one half of constant m times the square of the first derivative of the generalized coordinate.

Besides, capacitors can actually store energy and inductors cannot, at least not permanently, only temporarily in the magnetic field but things need to be in motion (changing).

my 2 cents
 
  • #5


The Lagrangian formulation of mechanics is a powerful tool for solving physical systems, including electrical circuits such as the LC circuit. In this approach, the Lagrangian is defined as the difference between the kinetic and potential energies of the system. In the case of the LC circuit, the inductor and capacitor both store energy, but in different forms.

The energy stored in the inductor is in the form of a magnetic field, while the energy stored in the capacitor is in the form of an electric field. In the Lagrangian formulation, the inductor energy is considered as kinetic energy because it is related to the rate of change of the charge, which is a measure of the motion of the charges in the circuit. This energy can be thought of as the "inertia" of the charges, as it resists changes in the charge flow.

On the other hand, the energy stored in the capacitor is considered as potential energy because it is related to the position of the charges in the circuit, specifically the separation of charges across the capacitor plates. This energy can be thought of as the "potential" for the charges to flow from one plate to the other.

Therefore, treating the inductor energy as kinetic and the capacitor energy as potential is consistent with the principles of the Lagrangian formulation and allows for a comprehensive understanding of the energy dynamics in the LC circuit.
 

1. What is a Lagrangian solution of an LC circuit?

The Lagrangian solution of an LC circuit is a mathematical method used to analyze the behavior of an electrical circuit that consists of an inductor (L) and a capacitor (C). It involves using the principles of Lagrangian mechanics to derive equations that describe the circuit’s energy and motion.

2. How does the Lagrangian solution differ from other methods of analyzing LC circuits?

The Lagrangian solution is a more general and elegant approach compared to other methods such as Kirchhoff’s laws or solving differential equations. It takes into account the system’s energy and potential, allowing for a deeper understanding of the circuit’s behavior.

3. What are the advantages of using the Lagrangian solution for LC circuits?

One of the main advantages of using the Lagrangian solution is that it can provide a complete analysis of the circuit’s behavior, including the effects of non-conservative forces such as resistance. It also allows for the calculation of the circuit’s total energy and its rate of change over time.

4. Are there any limitations to the Lagrangian solution of an LC circuit?

While the Lagrangian solution is a powerful tool for analyzing LC circuits, it does have some limitations. It assumes that the circuit is linear and time-invariant, and it does not take into account any non-idealities such as parasitic capacitance or inductance.

5. How can the Lagrangian solution be applied in practical situations?

The Lagrangian solution can be applied in practical situations to design and analyze complex electrical circuits, such as in power systems, telecommunications, and electronic devices. It can also be used to optimize circuit performance and troubleshoot any issues that may arise.

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