Talk:Lagrangian mechanics

Reviewer A

I have completed my review of the manuscript ``Lagrangian Mechanics by A. Bloch and D. Zenkov, which was submitted to Scholarpedia for online publication. My first impressions are that it is not clear what the purpose of the article is. While the manuscript is written from a mathematical perspective (understandable from the Authors' institutional affiliations), none of the physics applications of Lagrangian Mechanics, as well as the most important properties of Lagrangian systems with finite degrees of freedom, are discussed. In my opinion, the entire manuscript needs to be rearranged before it can rightfully be placed in the ``Physics section of Scholarpedia.

I would completely omit the section on Newton's Laws. (It simply must be assumed that these Laws are known by all potential readers.) First, I would instead begin with a brief review of the Calculus of Variations (e.g., Euler's First and Second equations as well as Calculus of Variations with constraints) and some simple classical problems (e.g., Fermat's Principle, the Brachistochrone problem, etc). Second, a brief historical survey of Principles of Least Action (from Maupertuis and Jacobi to Hamilton) would be presented. Third, single-particle Lagrangian mechanics would be introduced (with and without constraints) and the use of generalized configuration-space coordinates would be motivated with some examples (e.g., a particle constrained to move on a surface under the influence of a potential). Fourth, Noether's Theorem would be used to relate Lagrangian symmetries with conservation laws, and Routh's method could then be introduced to show how the derivation of a simple reduced Lagrangian is constructed. Fifth (and lastly), I would (perhaps) present a brief discussion of recent (and ongoing) work on geometric mechanics (e.g., action principles on Lie algebras, as applied to rigid-body mechanics).

Reviewer B

I'm sorry to say that, but I believe that, in spite of authors' efforts, the article in its present form is incomplete and must be largely expanded, at least in the directions that follows. If the article turns out to bee too long some parts can be put in separate appendices. I'll be happy to see the article again after a thorough revision: good work!

• 1) The role of d'Alambert principle of virtual works as the origin of the Lagrangian formalism in non-relativistic Lagrangian mechanics is totally minimized. The fact that the forces from holonomic constraints exerce no virtual work on a particle is a key physical point that leads to the validity of the lagrangian formalism. A discussion like that eg of Goldstein "Classical mechanics" section 1-4 is imperative.

• 2) The definition of holonomic constraint shoudl be given in the implicit form f(x,t)=0. Generalized coordinates should be defined as global solutions of these equations (when available). The use of lagrangian multiplier to deal with implicit constraint should be treated. The authors might reconsider to mention some important ideas about non-holonomic systems (or maybe write a separate article on that).

• 3) The formalism should be (trivially) extended to the case when constraints, Q and V are functions of t.

• 4) The notion of cyclic coordinates and the associated conserved quantities, including energy, should be introduced. The relation of conserved quantities and symmetries should be mentioned.

• 5) the fact the the lagrangian is not unique and defined up to total time-derivatives df(q,t)/dt shold be mentioned.

• 6) The case in which exists a generalized potential U such that Q=-d U/dt+d(dU/d qdot)/dt should be discussed and the key example of Lorentz force discussed.See Goldstein 1-5.

• 7) The question of gauge invariance in lagrangian formalism must be addressed, or at least the scholapredia article http://www.scholarpedia.org/article/Gauge_invariance should be referred to.

• 8) Friction should be discussed as an example of Q that does not admit U. See goldstein 1-5.

• 9) the extentions of the formalism to a relativistic test particle and a particle in curved space should be discussed, see eg the analog article in wikipedia http://en.wikipedia.org/wiki/Lagrangian and Goldstein 6-5.

• 10) few examples of lagrangians should be shown:armonic oscillator, relativistic particle in lorentz force, particle in curbed space, central potentials (and related cyclic coordinates).

• 11) the fact that Lagrangian formalism can be extended to continuous systems with infinite number of dof should be mentioned (this topics deserves a separate article)

other points

• 1) Lagrangian Mechanics is a fundamental invariant formulation of the fundamental laws of mechanics of particles and rigid bodies.

Invariant under what?. Which mechanics (classical, relativistic)?

• 2) Every particle continues in its state of rest or of uniform velocity in a straight line unless compelled to do otherwise by a force acting on it.

MISLEADING:The three laws hold only in an inertial system. The first law is the definition of inertial system. If the authors wants to cite all three Newton laws (Why?) they should explicitly mention the concept of inertial system.

• 3) the first set of equations above define position, or holonomic, constraints imposed on the system.

unclear which set: use eqn numbering/referencing

• 4) ...denotes a virtual displacement

virtual displacement in bold as definitions.

• 5) Then one can show (the infinitesimal form of the Lagrange--d'Alembert principle)

that missing

F_r in the eqn below is stricly speaking undefined.

why the principle is called Lagrange-d'alambert instead of d'alambert?

• 6) where T is the quadratic kinetic energy rewritten in terms of the variables

write T=1/2 m v^2

• 7) we define the Lagrangian to be

boldify lagrangian

• 8) We are given a real-valued function L(q^i,\dot{q}^i), called a Lagrangian.

the term lagrangian defined above; bold to be suppressed. Same consideration for kinetic energy and potential energy.

• 9) where the variation is over smooth curves in Q with fixed endpoints.

write explicitley q(a)=q_a q(b)=q_b fixed.

• 10) It should be mentioned that \int L dt is known as action and that what the authors calls Hamilton Principle (explain why?) is widely known as "Least action principle" (or more precisely stationary action principle). I've seen that an article "Least action principle" is scheduled on scholarpedia
• 11) the particular curve q(t) that is sought is a critical point of the quantity

critical point is not a definition; replace bold by italics.

• 12) A basic result of the calculus of variations is:

notrivial, citation needed

• 13) The proof uses the integration by parts and the boundary conditions.

the proof is easy to write and useful for understanding, authors should write it down.

• 14) In the presence of external forces, the equations of motion become

inthe presence of forces external to the system described by the lagrangian L...

• 15) These equations can be derived from a variational-like principle

use eqn reference

• 16) I would enjoy if the authors (or the editor) could provide a figure showing various trajectories at fixed boundary points (on which on minimize the action) and add a couple of references to introductory books more appropriate for students without a mathematical background (eg Goldstein).
• 17) The layout of the article should be improved. Some double white lines removed. Links to other articles in Scholarpedia introduced. Other articles have also useful see also and further reading sections.

Reviewer C:

There are two things that seemed missing from the article. I have addressed most of my concerns by modifying the article.

First: It is misleading to seem to imply that d'Alembert's principle is somehow necessary for the development of Lagrangian mechanics. They are logically independent.

Second: The authors chose to consider the case m < n, which implies a constrained system. The assumption that the forces of constraint do zero virtual work is essential to the derivation of the Lagrange equations for such systems and must at least be mentioned.

I assume that the authors of the article were under fairly severe constraints as to number of words allowed. They are to be complimented for doing a good job in a short space.