Relationship between Lagrangian and Energy

• theneedtoknow
In summary, the Lagrangian is a mathematical function that represents the total energy of a system in classical mechanics. It is directly related to energy through the Euler-Lagrange equation and plays a fundamental role in describing the motion and energy of a system. It differs from the Hamiltonian by considering kinetic and potential energies separately, and it can also be applied in other fields of physics such as quantum mechanics and electromagnetism.

theneedtoknow

I know that, for time-independent potentials, we have E=sum (Vi*partial dL/dVi) - L
What if one or more of the potentials are time-dependent?
Is the relationship between energy and the lagrangian then "total dE/dt = - partial dL/dt "?

Yes.

Thank you! Would you happen to know of any textbooks where this is derived? I have the Landau and Lifgarbagez "Mechanics" one and they do not do so.

1. What is the Lagrangian?

The Lagrangian is a mathematical function that represents the total energy of a system in classical mechanics. It takes into account the kinetic and potential energies of all the particles in the system.

2. How is the Lagrangian related to energy?

The Lagrangian is directly related to the energy of a system through the Euler-Lagrange equation. This equation states that the derivative of the Lagrangian with respect to a particle's position is equal to the negative of its force. By solving this equation, one can determine the motion and energy of a system.

3. What is the role of the Lagrangian in classical mechanics?

The Lagrangian is a fundamental concept in classical mechanics and is used to describe the motion and energy of a system. It allows for a more elegant and efficient approach to solving problems, as it takes into account all the forces acting on a system at once.

4. How does the Lagrangian differ from the Hamiltonian?

The Hamiltonian is another mathematical function used in classical mechanics to describe the energy of a system. While the Lagrangian considers the kinetic and potential energies separately, the Hamiltonian combines them into a single function. The two are closely related, as the Hamiltonian is equal to the sum of the Lagrangian and the rate of change of the Lagrangian with respect to time.

5. Can the Lagrangian be used in other fields of physics?

Yes, the Lagrangian can be applied to other areas of physics, such as quantum mechanics and electromagnetism. In these fields, it is used to describe the dynamics and energy of a system at the quantum level. It is also used in fields such as optics and fluid mechanics to describe the behavior of waves and fluids.