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Form of the Lagrangian

  1. Nov 23, 2015 #1
    I know this has been asked before: "Why is there a negative in the Lagrangian: L = T - V"

    I have read the answers and am not happy with them so I tried to formulate my own justification and now ask if anyone could comment on it?

    First, I am not happy with those who say "Because it works and F=ma comes out of it." I am just not happy with that, but I understand it and I suppose I can live with it.

    Second, I am not happy with those who write that it is a special case of higher physics (I forget where I read this, but I did not understand the higher physics involving gravitational potentials.... it was too advanced for me.

    Finally, I am not happy with the documents -- many referenced here in previous posts -- that discuss how a thrown ball "wants" to maximize height and minimize its speed. Before I continue, let me add that I DO understand those discussions... sort of... But I am left with asking: Why doesn't the ball want to stop and get a cheeseburger? (Basically, I am unhappy with the anthropomorphic answers on that level... )

    So...

    Could people advise me on my answer? And I am going to "frame" this answer under the umbrella of a simple falling particle.

    If I drop a ball, it will lose potential energy and gain kinetic energy. And there must be a continuous flow, with no loss of kinetic into potential. Now over the beginning and ending of the path, V (assuming the datum is the ground) becomes T. And these are large values and one should not expect THAT difference to be minimized.

    However, during the "incremental motion" along the path (inside the integral), V must be continuously converted into T. Thus, it makes sense, in the integrand, to minimize the difference between T and V: it is essential that along the path, loss in V continuously becomes gain in T.

    Does this make sense?

    If not... can I take it from the top and could someone tell me why the "ball WANTS to maximize its height and minimize its speed? What is the ball "thinking?" Or give me another reason for the form of the Lagrangian other than the trivially obvious "because it produces F=ma."
     
    Last edited: Nov 23, 2015
  2. jcsd
  3. Nov 23, 2015 #2

    MarcusAgrippa

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    Good question. In spite of the various rationalizations that you report and that I have seen, I personally don't know whether the question is even sensible. The Lagrangian is not uniquely defined. Many Lagrangians give rise to the same equations of motion. In the end, it is the equations of motion that contain the physics, not the Lagrangian function. I am inclined to think of a minimization principle as a convenient post- hoc rationalization without intrinsic physical significance but with fantastically powerful mathematical consequences. I went through a phase of trying to think of L as a kind of free energy, but that approach got me nowhere. Interestingly, Lagrange did not assign a physical significance to the Lagrangian function (as far as I am aware), but only to the equations of motion. Hamilton eventually realized that those equations could be derived from a variational principle an interpreted L dt as the action of Maupertuis.

    I am curious to see what others think.
     
  4. Nov 23, 2015 #3
    OK then...

    So when I apply Hamilton's Principle, and take the variation of essential displacements, I can construct the coupled equations of motion for a system of masses.

    I could also construct them by free body diagrams for each component part, but Hamilton's Principle with an appropriate Lagrangian, gives them... well... not as much "faster," but in a "cleaner" way and using generalized variables.

    And with YOUR response I feel I have taken a "baby step" toward FINALLY admitting (as you have said) "that the attempt to justify the FORM is not really a sensible thing to do." Rather, I REALLY SHOULD accept that the form provides the coupled system of equations. And be done with it.

    Could you comment on this? I mean I am ALMOST at the point of "forgoing" any attempt to justify the form of the Lagrangian. I suppose, first though, I need a better statement JUSTIFYING why justifying the form is NOT productive.
     
    Last edited: Nov 23, 2015
  5. Nov 23, 2015 #4

    MarcusAgrippa

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    My feeling is this - "feeling" because I am not sure that my justification is correct - if the form of the Lagrangian for a given system is not unique, does it make sense to look for an interpretation for the particular form of the Lagrangian that you have chosen to use? Why interpret one and not the others?
     
  6. Nov 23, 2015 #5
    OK; and with that, another baby step toward accepting that I have been spending unnecessary time.

    Yes, I can see your point here, too.

    So may I ask you (and forgive me for this pedagogical spin), what is the "learning structure" involved in all those "rationalizations" that have driven me off track? Why do people even write them, if they cannot even address "why the ball WANTS to do such such"

    I am really focused on your statement "am inclined to think of a minimization principle as a convenient post-hoc rationalization without intrinsic physical significance." I feel there is a truth here.

    But why have I never encountered this before? In all my other studies I have never come up against a wall like this; specifically: a formalization of a method that "works" in a grander sense and for which no justification is needed.

    Oh,I don't know what I am really asking. Maybe I have just been consumed thinking all those other "justifications" where justified.
     
  7. Nov 23, 2015 #6

    Dale

    Staff: Mentor

    I think you have come up against a wall like this before and just not realized it. Newton's laws are also a method that "works". The one and only justification for any method in physics is that it works. If it does not work thenmit is rejected no matter how elegant or appealing it is. If it does work then it is accepted no matter how unappealing or counter intuitive it might be.
     
  8. Nov 26, 2015 #7
    For a class of dynamical systems it can be proved that the lagrangian

    For a certain classes of dynamical systems you can go the other way around and prove the Hamilton's principle and the form of the lagrangian (up to a total derivative) .
     
  9. Nov 26, 2015 #8

    MarcusAgrippa

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    I suppose we always want to find reasons for why things are as they are. If you have only ever seen one Lagrangian for a dynamical system, and you are unaware that there may be many others that lead to the same equations of motion, you may be tempted to try to "understand" why the Lagrangian takes that particular form. After all, it really does look dangerously like a free energy! And that makes it very tempting to look for a deeper explanation! Even an anthropomorphic one, like "the ball wants to ..."

    Don't forget furthermore that one might still be infected with Aristotelian concepts of causality. One may then be tempted to look for a "formal cause" or a "teleological, or final cause". Tempting. If one can spin a vaguely credible tale, somehow we find that satisfying. A bit like a religion, really. Then all can pay lip service to the explanation, to a resounding "amen".

    I was taught mechanics from Hamilton's principle. Luckily, I had also previously had a course in Newtonian Mechanics. As a graduate student, I often marvelled how anyone could ever have arrived at such a procedure. After all, it is completely counter intuitive and esoteric as a foundational principle.

    After extensive reading, I gradually came to the realisation that Euler and Lagrange did not begin from a variational principle, but rather had looked for and had succeeded in finding a way to rewrite Newton's equations in coordinate independent form:
    [itex] \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}^i} \right) - \frac{\partial T}{\partial q^i}
    [/itex]
    is the i-th component of the configuration space acceleration in an arbitrary system of coordinates in the configuration space. So Newton 2 becomes
    [itex] \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}^i} \right) - \frac{\partial T}{\partial q^i} = Q_i
    [/itex]
    where Q_i is the i-th component of the generalised force that acts on the representative configuration point. In the special case where the generalised force is conservative, it can be expressed as the negative gradient of a potential V that is a function of the generalised coordinates only,
    [itex] \frac{d}{dt} \left( \frac{\partial T}{\partial \dot{q}^i} \right) - \frac{\partial T}{\partial q^i} = - \frac{\partial V}{\partial q^i}
    [/itex]
    leading to the familiar (but more special) form of Lagrange's equation - the less familiar form given above is the general form!

    Only much later did Hamilton devise Hamilton's principle which reduces Newtonian mechanics to a variational problem. The variational principle is thus, historically, an afterthought - albeit one with very powerful consequences. Note that there is not just one variational principle for mechanics. Another is Cartan's variational principle.

    From this, you can see that a variational formulation is best regarded as a post hoc rationalisation. This does not detract from its importance and power, but it puts it into its proper historical perspective.

    Why a variational principle? There are two strands to this answer. First, the variational principle has a long history in optics. It was devised by Heros of Alexandria to explain the laws of propagation of light, including free propagation and reflection. It was extended by Fermat to include Snell's law, and provided a theoretical derivation of Snell's law from a single optical principle. It thus was (and is) a very successful principle in optics.

    In Mechanics, the variational principle has a less distinguished history. It was claimed by Maupertuis that mechanical systems follow that path that requires least action. He claimed this for no better reason than that "God does everything economically". Maupertuis did not give a consistent definition of action. In fact, action was whatever he wanted it to be in any particular example, and he changed his definition with every application in order to force the principle to work. The principle of Maupertuis was thus frivolous, fatuous, useless and unfounded. However, Maupertuis was Euler's boss at the mathematical institute, and Euler did not want to cross him. So Euler found a way to define the "action" in such a way that a consistent definition could be used in a range of elementary examples. The number of examples was extended by Lagrange. It was not until Hamilton, however, that "action" was given a general definition, valid for all mechanical systems, and which worked every time. When the power of the variational principle was realised over the next century, it gained currency in mechanical theory and was advocated by some as more fundamental than the principles of Newtonian mechanics.

    It now looks as if any set of differential equations, total or partial, is susceptible to variational formulation. However, this is no more than a conjecture, and I know neither a proof of this conjecture, nor a disproof of it. The variational principle is therefore here to stay.

    I hope these random thoughts are helpful.
     
    Last edited: Nov 26, 2015
  10. Nov 26, 2015 #9
    If there are certain parts that seem unclear and need further elaboration, do let us know!
    Hamilton devised the principle of stationary action based on D'Alembert's principle. Equilibrium positions of mechanical systems occur for configurations where the potential energy is at a minimum - thereby resulting in no impressed forces on the system. Thus, the concept of equilibrium introduced by Newton, later took the form of a ##\textit{principle}##, which states that a system is in static stable equilibrium iff the virtual work of the impressed forces associated with an arbitrary variation of the generalised coordinates vanishes (remember, the potential is at a minimum at such points). D'Alembert's principle is basically an extension of this virtual work principle to ##"\textit{dynamical}"## equilibrium, as opposed to static equilibrium. In this principle, the total virtual work of the impressed forces ##\textit{and}## the inertial forces (the mass times acceleration part of Newton's eq), vanishes. This was the first reformulation of Newton's equations in terms of a principle. All "other" principles of analytical mechanics are in fact a reformulation of this principle. There is a problem however with this principle: although the virtual work of the impressed forces are integreable (i.e. an exact differential: ##\delta w## described by a scalar function), the virtual work of the inertial forces ##\bar{\delta w_i} ## is not, it is simply a differential form: ##\bar{\delta w_i} = \sum_j m_j \textbf{A}_j \cdot \delta\textbf{R}_j##. The key feature which Hamilton realised and the reason a minus sign appears in the Lagrangian function for "conservative" systems, is that ##\bar{\delta w_i}## can be transformed into an exact differential through an integration over time. Mathematically, the minus sign appears due to an integration by parts. Hamilton's principle and D'Alembert's principle are mathematically equivalent and their scopes are the same as long as the impressed forces can be derived from a scalar potential. If you would like the equations describing the steps I have outlined or a simple example which outlines this process and gives more of a "physical meaning" to this minus sign, let us know.
     
  11. Nov 28, 2015 #10
    PERFECT! Thank you.
     
  12. Nov 29, 2015 #11

    vanhees71

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    From the point of view of classical mechanics the principle of least action in its various forms is just a very elegant mathematical tool to describe the dynamics of the system. It's very powerful particularly because it opens the whole concept of group theoretical analysis in terms of Noether's theorem. From the point of view of physics symmetry principles are the most important heuristic ingredients of model building, because Noether's theorem works in both directions: From a Lie symmetry (i.e., a one-parameter Lie group of transformations on the phase-space variables that leave the variation of the action invariant) follows a conservation law. Also, if there is a conserved quantity it is the generator of a one-parameter Lie symmetry. Thus from the empirical knowledge of a conservation law you can constrain your action to those that obey the corresponding symmetry.

    Now you ask, why the usual form of the Lagrangian is ##L=T-V## (which is true for many Newtonian mechanical systems). The simple answer is that this is just the most simple form coming out when exploiting the full Galilei group, i.e., the continuous part of the full symmetry group of the Newtonian space-time model. This leads to pretty sharp constraints for the form of the action functional since it is a pretty large symmetry group (4 space-time translations, 3 rotations, 3 Galilei boosts).

    You can also look at the issue from the point of view that classical mechanics is an approximate formulation of quantum mechanics, if the state of the system is such that the action is very large compared to the modified Planck constant, i.e., ##A \gg \hbar##. This is most easily seen when using the path integral formulation of quantum theory. Then you find the classical approximation (and quantum corrections in terms of the ##\hbar## expansion of the quantum action) using the method of stationary phase to evaluate the path integral that describes the time evolution of the system (Green's functions). In the lowest order this leads to the classical action (generally in the Hamiltonian formulation with paths in phase space but often also to the Lagrangian version, because usually the Hamiltonian is of the form ##H=p^2/2m + V##, and you can do the momentum path integral exactly, which leads to the Lagrangian version of the path integral for paths in configuration space).

    Mathematically the stationary phase method in the path-integral formalism is equivalent to the WKB method (eikonal approximation) in wave mechanics, which in lowest order leads to the Hamilton-Jacobi partial differential equation for the classical action, which is just equivalent to the Hamilton principle of least action. In fact, Schrödinger took the opposite way when he discovered modern quantum theory in terms of wave mechanics: He considered the classical equations of motion of point particles as the eikonal approximation of a yet to determine wave equation, which then turned out to be, of course, the Schrödinger equation.
     
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