# What are Hamiltonian Mechanics?

• Lucretius
In summary, classical mechanics uses the Hamiltonian function, the Hamilton action, and the principle of least action to describe movement. Newtonian mechanics uses forces to describe movement, and Lagrangian mechanics uses energy to describe movement. Hamiltonian mechanics is an energy-based method that uses coordinates and momenta to describe movement.
Lucretius
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

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.

Then, do the harmonic oscillator.

These look useful:
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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## 1. What is the definition of Hamiltonian Mechanics?

Hamiltonian Mechanics is a mathematical framework used to study the dynamics of systems in classical mechanics. It was developed by Irish mathematician and physicist William Rowan Hamilton in the 19th century and is based on the principle of least action.

## 2. How does Hamiltonian Mechanics differ from Lagrangian Mechanics?

While both Hamiltonian Mechanics and Lagrangian Mechanics are used to study the dynamics of classical systems, they approach it from different perspectives. Hamiltonian Mechanics uses the concept of energy, while Lagrangian Mechanics uses the concept of generalized coordinates.

## 3. What is the role of Hamilton's equations in Hamiltonian Mechanics?

Hamilton's equations are a set of differential equations that describe the dynamics of a system in Hamiltonian Mechanics. They are derived from the Hamiltonian function, which represents the total energy of the system, and provide a complete description of the system's motion.

## 4. What are canonical transformations in Hamiltonian Mechanics?

Canonical transformations are transformations of coordinates and momenta in Hamiltonian Mechanics that preserve the form of Hamilton's equations. They are used to simplify the equations of motion and can also reveal symmetries in the system.

## 5. How is Hamiltonian Mechanics applied in real-world problems?

Hamiltonian Mechanics has numerous applications in various fields such as physics, engineering, and chemistry. It is used to study the motion of celestial bodies, analyze the behavior of mechanical systems, and understand the behavior of quantum systems. It also has applications in optics, fluid mechanics, and statistical mechanics.

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