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For a given Newtonian dynamical system, the Lagrangian function is
unique (up to a "total time derivative").
false |
It is a point of supprise to many that there may be a
number of NOT EQUIVALENT Lagrangians describing the same dynamical system.
EXAMPLE: the 2-dimensional oscillator has these two different Lagrangians
L = 1/2 ( x12 - x22) - 1/2 ( v12 - v22) The problem of existence and ambiguity of Lagrangian description is investigated as so-called the "inverse problem of variational calculus" since at least the 50s and is not resolved yet. | |||
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Hamilton (or Lagrange) equations are just alternative ways
of writing the Newton's equations of motion.
false | The choice of a particular Lagrangian brings to the description of a system some superstructure, not present in the Newtonian equations of motion. But the meaning of this superstructure lies beyond classical mechanics... | |||
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Once you have determined forces of a dynamical system,
you should be able --in principle-- to find its quantum description.
false | Between the Newtonian description of motion and its quantum description there is a Hamiltonian or Lagrangian ("No Lagrangian - no quantization!"). The choice of a particular Lagrangian (among many) determines the quantum picture of the system. The meaning of Lagrangian (or Hamiltonian) is rooted in the quantum nature of reality. | |||
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Maupertuis was a lunatic.
false | And this is why Maupertuis, commonly derided by his contemporaries for seeing in the principle of minimal action a divine elements was not entirely a lunatic. There is a (divinely guessed?) depth in bringing wave elements of optics into mechanics, as the quantum revolution showed 200 years later. | |||
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The difference between the Hamilton and the Lagrange equations
of motion is that the Hamilton equations are first order while the Lagrange
equations are second order.
false |
This is a popular misconception. Clearly, both Lagrangian and Hamilton
equations can be written as first-order equations of motion:
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Hamiltonian equations live on cotangent bundle
while Lagrange equations on a tangent bundle.
false | Another common misconception. You can have both equations defined on both (or any) manifolds. However, Hamilton's equations require a choice of an "observer" (trivialization of the fiber bundle of histories). | |||
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There is no Legendre transformation without a Lagrangian.
false | This is another misconception. Any dynamical system on a tangent space to a configuration space determines a map to the tangent space. Typical textbook-description of this map involves Lagrange function, which obfuscates a simple geometric nature of this map. | |||
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Hamilton-Jacobi equations concern canonical transformations.
false | The Hamilton-Jacobi equations are most often introduced (obscurely) as a result of "canonical transformation of coordinates." But in fact -- HJ equations concern existence of a Lagrangian submanifold in the phase space on which a "wave-phase" is determined. This is a "proto-Feynman" quantization of the system. | |||
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Hilbert's call to geometrize classical mechanics has found answer in
terms of symplectic geometry.
false | There are many alternative geometric descriptions beyond Lagrange or Hamilton's equations (symplectic geometry) that may be called "analytic description of motion." Among these exciting new propositions are: Lax equations (on a Lie algebra), bi-Hamiltonian systems, Nambu mechanics, to mention some most popular. Each of them raises new problems about quantization rules, integrability, etc. |
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