Comprehensive List of Mechanics Formulations

[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

 

[jedishrfu]

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:

• Newtonian mechanics
• Lagrangian mechanics
• Hamiltonian mechanics

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

• Celestial mechanics, relating to stars, planets and other celestial bodies
• Continuum mechanics, for materials modeled as a continuum, e.g., solids and fluids (i.e., liquids and gases).
• Relativistic mechanics (i.e. including the special and general theories of relativity), for bodies whose speed is close to the speed of light.
• Statistical mechanics, which provides a framework for relating the microscopic properties of individual atoms and molecules to the macroscopic or bulk thermodynamic properties of materials.

 

[Haborix]

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).

 

[Frabjous]

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

 

[jedishrfu]

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.

For other readers here, it's the box on the right side of the article NOT the article's table of contents.

 

[Frabjous]

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.

For other readers here, it's the box on the right side of the article NOT the article's table of contents.

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

 

[andresB]

Operational classical mechanics
https://iopscience.iop.org/article/10.1088/1751-8121/ac8f75

 

[andresB]

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.

For other readers here, it's the box on the right side of the article NOT the article's table of contents.

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.

 

[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.

 

[andresB]

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