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SeReNiTy
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Can someone point me in the right direction with the derivation number 2 from Chapter 2 (3rd edition) of Goldstein?
SeReNiTy said:The derivation is if the lagrangian contained velocity terms, derive what the conjugate momentum will be if the system is rotated by a angle in some direction.
The purpose of Derivation #2 in Goldstein Chapter 2 is to mathematically derive the equations of motion for a system with position-dependent forces. This derivation builds upon the concepts introduced in Derivation #1 and allows for a more general understanding of systems with varying forces.
The steps involved in Derivation #2 include setting up the Lagrangian for the system, applying the Euler-Lagrange equations, and solving for the equations of motion using the chain rule and the fundamental theorem of calculus.
Derivation #2 differs from Derivation #1 in that it takes into account position-dependent forces, whereas Derivation #1 only considers systems with constant forces. This allows for a more general understanding of systems with varying forces and leads to more complex equations of motion.
Derivation #2 has many real-world applications, including in the fields of mechanics, electromagnetism, and quantum mechanics. It can be used to model the motion of objects in varying gravitational fields, the behavior of electrical circuits with changing currents, and the quantum behavior of particles in potential wells.
You can use Derivation #2 to solve problems in your own research by applying the general equations of motion derived in this derivation to the specific system you are studying. This will allow you to accurately model and predict the behavior of your system and make informed conclusions about your research.