This chapter explores the public and political arenas within which the U.
Cleaning up the references[ edit ] I'm doing it now so please wait a while. It remains to trim the unfortunate walls of text but the content on constraints and degrees of freedom is important to include in this article, for the reader's referenceand add much more interesting examples: Landau and Lifshitz has loads including multiple coupled harmonic oscillators, anharmonic oscillators, driven-forced oscillators, rigid bodies, Hand and Finch also have a bunch of similar examples.
I intend to add them later. I will simplify by restricting 3d or lower dimensionsconsider most cases for one particle and generalize to many particles after, and streamline the text and notation more.
But the "pedanticness" of the validity of the results is very important to emphasize in detail which equations are more or less general than others, energy conservation, the nature of potential energy for different forces, etc. Also, the recently added topics like Newton's 2nd law in curvilinear coordinates, and the geodesic equation, are not irrelevant because they are exactly what is involved with Lagrangian mechanics, they do in fact provide the natural extension to general relativity, and citations have been added to support what has been said even more references will be added soon.
The mentioned examples can be done last thing. This article should concentrate on the non-relativistic formulation, and its too long to include relativistic and non-relativistic anyway. The discussion of how the system handles constraint forces comes before the definition of what a Lagrangian is?
Would a person who didn't already understand what a Lagrangian is understand the article? Seems to need a rewrite to have the introductory material first.
First the introduction shows examples where Lagrangian mechanics is useful, then the general definitions of the position and velocity vectors, and constraint equations and generalized coordinates, follow immediately to show how the examples fit in the with general definitions. This is the "introductory material first".
Then the Lagrangian is defined along with the definitions of kinetic and potential energy, and in painful detail the terms "explicitly dependent on time" etc.
Then are the equations of motion, followed by the transition from Newtonian to Lagrangian mechanics, and in the process the origins and validity of the equations are shown why they are true and where they come from. Then there are properties of the EL equations relevant to mechanics.
Then two coordinate examples, and two more detailed mechanical examples. Finally there are applications of the theory in other contexts.
So I would say this is very much in order, because there is motivation first thing for why and where the theory is even useful, and the definitions are introduced as needed and given in extremely explicit terms. Do you think it is bad to motivate with examples first?
I was suspecting people would complain for providing no motivational introduction, so put it first, followed by the general definitions of coordinates and velocities and constraints and the Lagrangian before the equations of motion. You don't need the definition of a Lagrangian to understand the examples, nor the ideas of constraints.
I'll wait for at least your reply before I try once more to rewrite the sections "in order". Does that sound better?
It is not widely stated that Hamilton's principle is a variational principle only with holonomic constraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving d'Alembert principle of virtual work.
Actually, Hamilton's principle does work for certain? We shouldn't go into too much depth in this article, but it should at least say Lagrange's equations of the first kind can be formulated with non-holonomic constraints, and make amendments elsewhere throughout where it says Lagrangian mechanics "only works for holonomic constraints".
The main article Hamilton's principle should contain the formalism with holonomic and nonholonomic constraints which it currently does not. As always, a good source for all this is Goldstein. I'll get back to this soon.
The generalized coordinates article has examples which reflect better the applications of D'Alembert's principle and Lagrangian mechanics.
This article has an introductory section motivating the use of generalized coordinates and how constraint equations are formulated, first with examples then generally for holonomic constraints. Maybe we could move the content in Lagrangian mechanics Introduction to somewhere in generalized coordinates, move the examples in Generalized coordinates Examples to the examples section in this article.
It is explained why in the article. I have reverted both edits. Let's see what user: AHusain has to say about this exercise 3.
That gives an example of how you can account for such forces in Lagrangian mechanics as well by using a judicious change of variables. So it is not really less general in the way described. The not introducing new physics statement is strictly true but not morally true because it does give the deformation theory for the quantization by the hessian at the critical point and not just the equation of motion the critical point.
So your removal of that was wrong.The aim of this chapter is to discuss mixed formulation and constraints.
The set of differential equations from which the discretization process is started determines whether the formulation is referred to as mixed or as irreducible.
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