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We usually discuss the ergodocity of isolated systems, which means that total energy is constant. Consequently there is no point in even mentioning those parts of phase space with the wrong energy. The question therefore is whether a system (with a given initial point in phase space) will within finite time come arbitrarily close to every point in the ...


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The canonical (total) momentum is the sum of the kinetic (mechanical) momentum and the potential momentum. Potential momentum occurs only if the potential energy depends explicitly on velocity.


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Yes lagrangians and hamiltonians are indeed used by engineers. To be precise, used by some types of engineers like aeronautical engineers, aerodynamics etc.. For example: http://www.osti.gov/eprints/topicpages/documents/starturl/47/566.html As far as i know electrical engineers dont use the lagrangian nor hamiltonian forms of mechanics nor ...


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I'm a electrical engineer, and have never used either one in over 30 years of designing circuits. I vaguely remember that we went over them briefly in school, but since I haven't used them (knowingly) since, I can't tell you what the physical meaning of either is, which of course perpetuates the fact that I'm not going to use them.


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In civil engineering they use it for structures, and strength of materials in the elastic realm. It goes by the name of the enegy method. Google books might give an indication. Some authors are Beer and the mechanical engineer Stephen Timoshenko. This is for some what "static" indeterminant structures. So, there is no time element. But, I am sure it ...


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The question is what do we need the matter content of the universe for. As I understand it, in the usual case we want to find the conserved quantity associated with a certain conserved current gained by the projection of the energy-momentum tensor into a Killing vector, as for example in the paper by Abott and Deser. The requirement of asymptotical ...


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I don't know where you're getting those $m$s from, or what substitution you're making. The appropriate substitution to perform is $$ p = \frac{\partial L}{\partial \dot{q}} = \omega \dot{q}. $$ If you do this, then the hamiltonian becomes $$ H = p\dot{q} - L = \frac{p^2}{\omega} - \frac{p^2}{2\omega} + \frac{1}{2} \omega q^2 = \frac{1}{2\omega}\left(p^2 + ...


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Richens & Berry [Physica 1D, 495-512 (1981)] give beautiful examples of such systems (with phase-space that is a surface of genus > 1), which they call pseudointegrable; their examples are invariant manifolds of billiards shaped as polygons with rational angles. These systems are interesting because there are two constants of motion, so invariant ...


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Calculating a Hamiltonian from a Lagrangian leads us to a new quantity, which is a function of coordinates and momenta. If you simply substitute velocity as a function of momentum into the Lagrangian, then the "Hamiltonian" will depend on velocity implicitly. This is not what we want. Let me show you an example to explain why Hamiltonian (the usual one) ...


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The key point in all of this is that general relativity is a gauge theory, and, as the saying goes, "the gauge always hits twice" (apparently attributed to Claudio Teitelboim). What this means is that (1) you have an arbitrary freedom in defining your evolution, corresponding to the ability to make gauge transformations, and (2) some of the evolution ...


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It is interesting to look at a linearized version of gravity, with $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$ If you choose the Lorentz gauge : $$\partial^\mu \bar h_{\mu\nu}=0 \quad\quad \bar h_{\mu\nu} = h_{\mu\nu} - \frac{1}{2} h^i_i \,\eta_{\mu\nu} \tag{0}$$ the equations of movement in the vaccuum are simply : $$\square \bar h_{\mu\nu}=0 \tag{1}$$ ...


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You don't need to express $t$ in terms of $z$ and $p$, explicit time dependence is permissible in the Hamiltonian. (It would not even be possible without restoring to $\dot z$.) In the other question it was not mentioned because it was not needed, the Lagrangian was time-symmetric and consequently so was the Hamiltonian. This is very often the case so many ...


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Well, it is simple. If $\omega_x/\omega_y$ is irrational, then the evolution visits the neighborhood of any allowed point in the phase space arbitrarily closely. This is pretty much a more general form of the claim that $\cos kN$ for $k$ irrational and $N\in{\mathbb Z}$ may belong to any interval $R\pm \epsilon$ for any $-1\lt R\lt 1$ and arbitrarily small ...


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Check out V.I. Arnold's Mathematical Methods of Classical Mechanics. This book is pretty terse and can have hard to follow notation. However, it is rigorous and contains mathematical explanations and proof of a wide array of topics in mechanics. It is also filled with very interesting examples. He introduces the concepts needed from differential geometry; ...


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Noether's theorem in Hamiltonian mechanics is saying the same thing as Noether's theorem in the Lagrangian setting, under the Legendre transform. A Hamiltonian system is a triple $(M,\omega, H)$ where $(M,\omega)$ is a symplectic manifold and $H$ is the Hamiltonian. You define a continuous symmetry in the Hamiltonian setting to be a vector field $V$ that ...



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