# Comprehensive List of Mechanics Formulations

• I
• Al-Layth
In summary, the various formulations of classical mechanics include the Newtonian, the Lagrangian, and the Hamiltonian. The latter is the most mathematically sophisticated and is used when dealing with large systems.
Al-Layth
TL;DR Summary
Is there a comprehensive list of formulations of classical mechanics?
beyond
1.) the Newtonian formulation
2.) The lagrangian formulation
3.) The Hamiltonian formulation

What other formulations are there

Doesn't seem like there are any others at the moment:

https://en.wikipedia.org/wiki/Classical_mechanics

## Branches​

Classical mechanics was traditionally divided into three main branches:

• Statics, the study of equilibrium and its relation to forces
• Dynamics, the study of motion and its relation to forces
• Kinematics, dealing with the implications of observed motions without regard for circumstances causing them

Another division is based on the choice of mathematical formalism:

Alternatively, a division can be made by region of application:

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DeBangis21 and Al-Layth
D'Alembert's Principle might be worth noting separately even though it fits under the Newtonian formulation. And within the Lagrangian and Hamiltonian formulations there are varying degrees of mathematical sophistication (e.g. symplectic manifolds and symplectomorphisms versus phase space and canonical transformations).

jedishrfu
Wikipedia lists
Newton‘s Laws of Motion
Analytical Mechanics
Lagrangian Mechanics
Hamiltonian Mechanics
Routhian Mechanics
Hamilton-Jacobi Equation
Appell’s Equation of Motion
Koopman- von Neumann Mechanics
https://en.wikipedia.org/wiki/Classical_mechanics (look under formulations in the box)

I am not sure if Geometric Mechanics would count as an additional area

Interesting catch, I posted the branches portion of the wiki article, and it didn't reference those other formulations perhaps because they are a mix of Lagrangian and Hamiltonian formulations. I don't know.

jedishrfu said:
Interesting catch, I posted the branches portion of the wiki article, and it didn't reference those other formulations perhaps because they are a mix of Lagrangian and Hamiltonian formulations. I don't know.

It is unusual formatting. I was looking for Routhian and the list was opened on that page.

jedishrfu
jedishrfu said:
Interesting catch, I posted the branches portion of the wiki article, and it didn't reference those other formulations perhaps because they are a mix of Lagrangian and Hamiltonian formulations. I don't know.

The Routhian is indeed a mix between Lagrangian and Hamiltonian formalisms.
The Koopman-von Neumann stems froms the Liouville equation from Hamiltonian mechanics.
I suppose the term analytical mechanics emcompases stuff outside Lagrangian or Hamiltonian mechanics, like the Gauss and Jourdain principles.
The Gibbs-Apell approach seems more general than the Lagrangian and Hamiltonian approach, since Gibbs-Apell covers non-linear non-holonomic constraints.

vanhees71 and jedishrfu
I was blown away by the Lagrangian when I first learned it.

It has a kind Taoist philosophical bent to it in that Least Action is something like wu-wei (do nothing unnecessary, move like water...) At the time, my profs were musing about connections between eastern mysticism, the Yin-Yang and how it related to the standard model.

These were musings only as no physicist would ever digress into fields beyond reality. I think it was due to books like the Dancing Wu Li Masters by Gary Zukav that were talked about in Physics Today that fueled their interest.

Robert H. March, Professor of Physics at the University of Wisconsin, wrote in Physics Today in August 1979, "Dealing with general relativity [Zukav] manages to convey the profound mental shift required to reduce physics to geometry. This is a neat trick, considering that he addresses an audience familiar with neither physics nor non-Euclidian geometry."

https://en.wikipedia.org/wiki/Gary_Zukav

Hamiltonian formulation was taught because of its connection to Quantum Mechanics but to me was interesting because it was tied more to how one might program a computer to do a physics simulation with first order equations.

I guess it was our loss that we never covered the other formulations.

jedishrfu said:
I was blown away by the Lagrangian when I first learned it.

It has a kind Taoist philosophical bent to it in that Least Action is something like wu-wei (do nothing unnecessary, move like water...)

Notice that the Hamilton princple only has a variational interpretation when we consider Holonomic system. While there is a Hamilton principle for non-holonomic cases, it can't be interpreted as the variation of an action ##\delta S=0##

On the other hand, I will shamelessly take the opportunity to promote my own work on the variational principles of mechanics
https://iopscience.iop.org/article/10.1088/1751-8121/ac2321/pdf
https://arxiv.org/pdf/2107.03982.pdf

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## 1. What is a comprehensive list of mechanics formulations?

A comprehensive list of mechanics formulations is a compilation of all the mathematical equations and principles that describe the behavior of physical systems in motion. It includes topics such as classical mechanics, quantum mechanics, and fluid mechanics.

## 2. Why is a comprehensive list of mechanics formulations important?

Having a comprehensive list of mechanics formulations is important because it provides a systematic and organized approach to understanding and analyzing the behavior of physical systems. It also serves as a reference for engineers and scientists in designing and predicting the performance of various mechanical systems.

## 3. How is a comprehensive list of mechanics formulations used in research?

In research, a comprehensive list of mechanics formulations is used as a foundation for developing new theories and models to explain the behavior of physical systems. It is also used to validate experimental results and make predictions about the behavior of new systems.

## 4. Are there any limitations to a comprehensive list of mechanics formulations?

Yes, there are limitations to a comprehensive list of mechanics formulations. It is based on simplified assumptions and may not accurately describe the behavior of highly complex systems. It also does not account for factors such as friction, air resistance, and other external forces that may affect the motion of objects.

## 5. How can I access a comprehensive list of mechanics formulations?

A comprehensive list of mechanics formulations can be found in various textbooks, journals, and online resources. It is also commonly taught in undergraduate and graduate level courses in physics and engineering. Additionally, many software programs and calculators have built-in mechanics formulations for easy access and use.

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