Table of Contents
This tutorial provides a complete, step-by-step derivation of the Lorentz force law starting from the Lagrangian. The tutorial includes the derivation of key vector identities and explicitly details each term involved in the process. The goal is to make the content understandable even for someone new to the subject.
Step 1: Define the Lagrangian
The Lagrangian is a function that describes the dynamics of a system. For a charged particle of mass and charge moving in electromagnetic fields described by a vector potential and a scalar potential , the Lagrangian is defined as:
Here, is the velocity of the particle, and the dot represents a time derivative.
Step 2: Compute
To proceed, we need to find the partial derivative of with respect to . This derivative is obtained as follows:
Step 3: Derive the Vector Identity
Before diving into the next steps, let’s derive a key vector identity that will be used later. The identity is:
The derivation of this identity involves using the definitions of gradient, divergence, and curl, along with the product rule for derivatives. Due to its complexity, it’s typically proven using tensor notation or by working through each Cartesian component.
Step 4: Compute with Explicit Derivation of Terms
Now, we’ll find the partial derivative of with respect to . This involves differentiating the terms and .
Derivation of the Term
The term differentiates to when taking the gradient with respect to .
Derivation of the Last Two Terms from
Using the vector identity derived in Step 3, the gradient of becomes:
These two terms will be part of .
Final Expression for
Combining these terms, we get:
Step 5: Euler-Lagrange Equation and Time Derivative
The Euler-Lagrange equation states:
To apply this equation, we need to find the time derivative of , which is:
The total time derivative includes both explicit and implicit time dependencies:
Step 6: Substitute into Euler-Lagrange Equation and Simplify
Having all the necessary derivatives and expressions at hand, we can now substitute these into the Euler-Lagrange equation:
Substituting these into the Euler-Lagrange equation, we get:
Simplifying, we find:
Finally, using and , we arrive at the Lorentz force law:
This concludes the comprehensive tutorial on deriving the Lorentz force law from the Lagrangian. The tutorial aimed to be as detailed as possible, explicitly showing the derivation of each term and equation involved. I hope you find this tutorial complete and informative.
Derivation Using Limits
The Definition of the Total Derivative
The total derivative of a vector field is defined by the limit:
Breaking Down the Limit Expression
We can express using a Taylor series expansion around :
Now, we can substitute this expansion back into the limit expression for :
Factoring out in the numerator, we get:
Taking the Limit
As approaches zero, approaches , the velocity of the particle. So, we have:
This gives us the expression for the total time derivative , which includes both the explicit and implicit time dependencies.
Certainly, demonstrating the vector identity through a concrete 3D example can offer a tangible way to grasp its intricacies. The vector identity we’re interested in is:
For simplicity, let’s consider and as 3D vectors defined in Cartesian coordinates :
Tutorial: Verifying the Vector Identity in 3D
Step 1: Compute
The dot product is given by:
Step 2: Compute
The gradient of with respect to is:
This results in a vector with components:
Step 3: Compute and
The term can be written as:
After performing the dot product, we get:
A similar calculation can be done for .
Step 4: Compute and
Computing the curl and yields vectors in components. The cross product can then be computed term-by-term in a straightforward manner.