Which mechanics is best?

Question

Is there any specific reason for using Newton mechanics instead of Hamiltonian or any other theory of mechanics.

I am asking just out of curiosity (I know nothing about Hamiltonian mechanics)

Answer 1

Newtonian mechanics is good for representing physical systems in terms of the most basic elements. You have Forces, masses, velocities and accelerations having a relationship represented by an equation of motion. They imply properties of how energy changes within the system. The equations of motion can be very difficult to derive though. For example, suppose you have two pendulums attached to each other. Where the rod of one pivots around the bob of another. What are the forces involved? Then even simple systems represented in non-cartesian coordinates can get really complicated in the Newtonian representation.

Lagrangian Mechanics has some advantages. That pendulum problem is difficult in terms of forces, but the energies or fairly easy to calculate and The Laplacian of a system represents the mechanics of a system in terms of its Action, a relationship between kinetic and potential energy. The energies usually take on much simpler forms in various coordinate systems. Once one has the Lagrangian, the Euler Lagrange equations readily lead to equations of motion.

Hamiltonian Mechanics has a lot of the same benefits of Lagrangian Mechanics. In fact, the Hamiltonian is derived from the Lagrangian. Hamilton's approach splits the second order differential equations from the Laplacian treatment to twice as many first order differential equations. It's much easier to use numerical solvers on first order than second order equations. Results are also usually more accurate. So Hamilton is the way to go if you are making a physics engine.