# Magnetism and lagrange formulation of mechanics

• Nick R
In summary, the conversation discusses the role of magnetic forces in conservative systems and how they can be accounted for in the Lagrangian formulation of mechanics. It is explained that the Euler-Lagrange equation only requires knowledge of the potential energy and kinetic energy of the system. The example of a steady current and a charged particle experiencing a central force is used to demonstrate this concept. It is noted that the Lagrangian may not contain information relevant to the magnetic force, but it is still conserved in this system. The potential energy is also mentioned to be velocity dependent in this situation.
Nick R
Hello, I am aware that magnetic forces can do no work.

I am also aware that, in a conservative system, equations of motion that minimize the "action" (which are the true equations of motion) can be found with the euler-lagrange equation. The only information the euler-lagrange equation needs about the system is the potential energy as a function of position and kinetic energy as a function of the first time derivative of position.

So take a system in which there is a steady current (in some direction) flowing to infinity along some axis, and a charged particle moving orthonally to the steady current.

The charged particle will experience a central force (magnetic) that will cause it to orbit the current at some radius.

So my question is, how is it possible for the lagrangian formuation of mechanics to account for this simple sitauation? The magnetic force can do no work, so it can't affect the potential energy field (right?), and the kinetic energy is always just (1/2)mv^2. It would seem that the lagrangian can contain no information relevant to the magnetic force (the lagrangian being defined as T - U ; which is the only item having any information relevant to the physical situation in the euler-lagrange equation).

I would like to address your question about how the Lagrange formulation of mechanics can account for the interaction between a charged particle and a steady current. While it is true that magnetic forces do no work, they still play a crucial role in the dynamics of a system. The Lagrange formulation takes into account all the forces acting on a system, including the magnetic force.

In this case, the Lagrangian would include the kinetic energy of the charged particle as well as the potential energy due to its interaction with the magnetic field generated by the steady current. This potential energy would depend on the position of the charged particle and the strength of the magnetic field.

The Euler-Lagrange equation takes into account all of these forces and their corresponding energies, and finds the path that minimizes the action (or energy) of the system. In this case, the path would be an orbit around the current, as you mentioned.

It is important to note that the Lagrangian formulation is a powerful tool in mechanics because it allows us to consider all types of forces, including non-conservative ones like magnetic forces. The Lagrangian contains all the relevant information about the system and allows us to find the true equations of motion.

I hope this helps clarify the role of the Lagrangian formulation in accounting for the magnetic force in this simple situation. Let me know if you have any further questions or concerns.

## 1. What is magnetism and how does it work?

Magnetism is a phenomenon where certain materials, such as iron, nickel, and cobalt, can attract or repel each other. This is due to the presence of magnetic fields, which are created by the movement of electric charges. In simpler terms, magnetism is the force that causes magnets to stick together or push apart.

## 2. How is magnetism related to electricity?

Magnetism and electricity are closely related as they are both manifestations of the electromagnetic force. When an electric current flows through a wire, it creates a magnetic field around the wire. Similarly, when a magnet moves near a wire, it creates an electric current in the wire. This relationship is known as electromagnetism and is the basis for many modern technologies such as electric motors and generators.

## 3. What is Lagrange formulation of mechanics?

The Lagrange formulation of mechanics, also known as the Lagrangian mechanics, is a mathematical framework used to describe the motion of objects in a system. It is based on the principle of least action, which states that the path an object takes between two points is the one that minimizes the action, a measure of energy over time. This formulation is widely used in various fields of physics, including classical mechanics and quantum mechanics.

## 4. How is the Lagrange formulation of mechanics applied to magnetism?

In the Lagrange formulation of mechanics, the equations of motion are derived from a single function, called the Lagrangian, instead of using multiple equations. This makes it a useful tool for analyzing complex systems, such as those involving magnetism. By considering the potential and kinetic energy of the magnetic field, the Lagrangian can be used to derive the equations of motion for magnetic systems.

## 5. What are some real-life applications of magnetism and the Lagrange formulation of mechanics?

Magnetism and the Lagrange formulation of mechanics have numerous real-life applications. Some examples include the use of magnetic fields in MRI machines for medical imaging, the use of electric motors in cars and appliances, and the use of Lagrangian mechanics in spacecraft trajectory planning. Additionally, understanding magnetism and the Lagrange formulation of mechanics is crucial in the development of new technologies and advancements in fields such as materials science and engineering.

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