- #1
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- 15
\displaystyle R_{\mu v} - \frac{1}{2} R g_{\mu v} + \Lambda g_{\mu v} = \frac{8 \pi G}{c^4} T_{\mu v}
\displaystyle i \hbar\frac{\partial \psi}
{\partial t}=
\frac{-\hbar^2}{2m}
\left(\frac{\partial^2}{\partial x^2}
+ \frac{\partial^2}{\partial y^2}
+ \frac{\partial^2}{\partial z^2}
\right) \psi + V \psi.
What do you mean by required? If you are doing an exam you might speak of requirements. Otherwise what is "required" is just that you convince yourself that it is true. Note that there is no need to pull the metrics out of the ##F_{\alpha\beta}## in your second term. The indices will be raised by the metrics from the ##F^{\alpha\beta}## once you pull them out of the derivative.grzz said:Is this working required for a beginner or is it to be left out?
Orodruin said:What do you mean by required?
The E-L (Euler-Lagrange) equations for EM (electromagnetism) are a set of partial differential equations that describe the behavior of electric and magnetic fields in space. They are derived from the Lagrangian density, which is a measure of the energy of a system.
The E-L equations are used to determine the equations of motion for electromagnetic systems. They are fundamental in understanding how electric and magnetic fields interact with each other and with charged particles. They are also used in the development of electromagnetic theories and in the design of electromagnetic devices.
The derivation of the E-L equations in EM involves applying the principle of least action to the Lagrangian density, which represents the energy of the system. This involves finding the stationary points of the action integral, which leads to the E-L equations. This process can be complex and requires a strong understanding of mathematical concepts such as calculus and variational calculus.
The E-L equations have a wide range of applications in the field of electromagnetism. They are used in the development of electromagnetic theories, such as Maxwell's equations and the theory of relativity. They are also essential in the design and analysis of electromagnetic devices, such as antennas, motors, and generators.
The E-L equations are closely related to other equations in EM, such as Maxwell's equations and the Lorentz force law. In fact, Maxwell's equations can be derived from the E-L equations and the Lorentz force law can be derived from the E-L equations and the conservation of energy and momentum principles. Understanding the relationships between these equations is crucial in developing a comprehensive understanding of electromagnetism.