For the 300-400 page version see: "Classical Mechanics" by Goldstein.
I highly recommended this book. The first four chapters are a solid treatment of everything you learned in first year physics. For example, you will understand that the angular velocity \omega is actually \vec{\omega}.
Okay, so the gist of the whole Hamiltonian/Lagrangian is you we can solve problems by using energy calculations. The //Hamiltonian// describes the total energy in a system H = K + V. The Lagrangian is a bit f-up because, apparently, all the information you will ever need about the system can also be computed[1] from the Lagrangian L = K - V. The relation between H and L is called the Legendre transformation.
But the fun doesn't stop there. We have three different ways to solve physics problems until now (1) Newton (dynamics->a->kinematics), (2) Lagrangian + L-eqns, and (3) Hamiltonian + H-eqns. You would think physicists would stop at this point. Be like "OK we got three now, done!", but no they thought of an even more general way to think about the world.
The Hamilton-Jacobi equation is the final piece of the classical mechanics puzzle. In the last three chapters of Goldstein, you will learn about the connection between the Hamilton-Jacobi and Schrodinger's equation. Essentially, if you take the limit $\hbar \to 0$ in the main equation of QM, it simplifies to the Hamilton-Jacobi equation. Read JJ.Sakurai's ``Modern QM'' to continue.
Comments
For the 300-400 page version see: "Classical Mechanics" by Goldstein. I highly recommended this book. The first four chapters are a solid treatment of everything you learned in first year physics. For example, you will understand that the angular velocity \omega is actually \vec{\omega}.
Okay, so the gist of the whole Hamiltonian/Lagrangian is you we can solve problems by using energy calculations. The //Hamiltonian// describes the total energy in a system H = K + V. The Lagrangian is a bit f-up because, apparently, all the information you will ever need about the system can also be computed[1] from the Lagrangian L = K - V. The relation between H and L is called the Legendre transformation.
But the fun doesn't stop there. We have three different ways to solve physics problems until now (1) Newton (dynamics->a->kinematics), (2) Lagrangian + L-eqns, and (3) Hamiltonian + H-eqns. You would think physicists would stop at this point. Be like "OK we got three now, done!", but no they thought of an even more general way to think about the world.
The Hamilton-Jacobi equation is the final piece of the classical mechanics puzzle. In the last three chapters of Goldstein, you will learn about the connection between the Hamilton-Jacobi and Schrodinger's equation. Essentially, if you take the limit $\hbar \to 0$ in the main equation of QM, it simplifies to the Hamilton-Jacobi equation. Read JJ.Sakurai's ``Modern QM'' to continue.
[1] http://en.wikipedia.org/wiki/Hamiltonian_mechanics#Calculati...
[2] http://en.wikipedia.org/wiki/Hamilton%E2%80%93Jacobi_equatio...