What are Hamiltonian Mechanics?

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Hamiltonian Mechanics is a formulation of Classical Mechanics that utilizes the Hamilton function and the principle of least action, focusing on energy rather than forces. It leads to a system of 2n first-order differential equations based on coordinates and momenta, contrasting with Newtonian and Lagrangian mechanics. Practical examples include analyzing a free particle and a harmonic oscillator. This approach can simplify the solution of many physics problems that are complex under Newtonian mechanics. Understanding Hamiltonian Mechanics opens up new perspectives in the study of physics.
Lucretius
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Can anyone give me a basic definition of Hamiltonian Mechanics without all the fancy mathematics, and perhaps could supply a few examples as to this? I am trying to make sense of this, but everywhere I go, I run into very large mathematical equations and no defintions I can understand.

Thanks,

Lucretius
 
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Classical Mechanics (CM,for short) has 3 formulations,one of which being the Hamiltonian one.So the definition could be:the formulation of Classical Mechanics based on:
a)The Hamilton function.
b)The Hamilton action.
c)The Principle of Least Action applied to th functional at b).


Daniel.
 
Newtonian Mechanics: based on forces; leads to a second-order differential equation for position.
Lagrangian Mechanics: an energy-method based on n coordinates and their velocities; leads to a system of n second-order differential equations.
Hamiltonian Mechanics: an energy-based method based on n coordinates and their momenta; leads to a system of 2n first-order differential equations.

Start with the free particle.
Then, do the harmonic oscillator.

These look useful:
http://academics.smcvt.edu/abrizard/Classical_Mechanics/Notes.pdf
http://core.ecu.edu/phys/flurchickk/Classes/PH4226/Section6/Index6.html (colorful)
 
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you might also try writing the Lagragrian and Hamiltonian equation equations and solving them for a particle involved in free fall.
F=mg choosing one dimensional motion and no air resistance of course.

You might ask yourself, why would anyone want to do physics this way instead of using Newtonian mechanics?

Well believe it or not many more problems in physics are easier and only possible to solve using these formulations!

HAVE FUN! You are no longer talking baby steps with Physics! You are entering Puberty!

Good Luck!

I am glad I do not have to grind those equations anymore!

love and peace,
and,
peace and love,
(kirk) kirk gregory czuhai
http://www.altelco.net/~lovekgc/kirksresume.htm
 
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