# Tag Info

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I'll write here a list of my personal favorites plus some commonly used books. I wouldn't be surprised if your teacher chose either one of the books below as a textbook: i) Mechanics, the first volume of the Landau course on Theoretical Physics; ii) Goldstein's book "Classical Mechanics"; iii) Taylor's book "Classical Mechanics"; iv) Marion's book ...

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The basic idea is the following. For the shake of simplicity, I henceforth assume that every function does not depend explicitly on time (with a little effort, everything could be generalized dealing with a suitable fiber bundle over the axis of time whose fibers are spaces of phases at time $t$). On a symplectic 2n dimensional manifold (a space of ...

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Here is one line of motivation: On one hand, in the Lagrangian formalism, the Lagrangian energy function $$\tag{1} h(q,v,t)~:=~v^i \frac{\partial L(q,v,t)}{\partial v^i}- L(q,v,t)$$ is defined as the Noether charge for time translations. Noether's theorem states that if the Lagrangian is invariant under time translations, which implies that ...

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To start things off I'd say that noting the $L_z$ component is conserved seems to mean pretty much nothing, since you're considering the motion as restricted to the $\mathcal{X}\mathcal{Y}$ plane. If you had assumed the motion along the $\mathcal{Z}$ axis to be possible, then we'd be talking about the spherical double pendulum instead of the planar one ...

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The problem here is how to quantize systems whose classical hamiltonian involves factors of the form (for example) $p^nx^m$, because these cannot be unambigously represented in a formalism where $p$ and $x$ do not commute. As such there are many alternatives (all of which are classically equivalent) but only one is quantum-mechanicaly relevant. In most ...

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The phrase "the function is spherically simmetrical" means that, if $G$ is an orthogonal transformation (that sends spheres into themselves), then $$f(G\mathbf r, G\mathbf p,t)=f(\mathbf r , \mathbf p, t).$$ If you know $\mathbf r^2$, $\mathbf p^2$, $\mathbf r \cdot \mathbf p$ you can calculate $f$ by taking an orthogonal transformation which maps $\mathbf ... 2 Hamiltonian formalism won't help so much because the problem is dissipative. You can solve the homogeneous linear diff. equation with constant coefficients by supposing$x=C e^{\lambda t}$. You will get complex solutions, but you should be able to add them to get a real solution. Now you should have$x(t)$with two arbitrary constants determined by initial ... 2 I don't know how elementary you consider a simple position dependent mass, but due to ordering ambiguity in the kinetic term$\hat{p}^2/2m(\hat{r})$such a system will have a quantum Hamiltonian different from the classical one. For example: Analytic results in the position-dependent mass Schrodinger problem Position-dependent effective masses in ... 1 I) In general, for a given choice of boundary conditions, it is important to adjust the action with compatible boundary terms/total divergence terms in order to ensure the existence of the variational/functional derivative. As OP observes, the problem is (when deriving the Euler-Lagrange expression) that the usual integration by part argument fails if the ... 1 In this answer we will consider a Lie algebra$L$(rather than a Lie group). Then: If$M$is a manifold, let there be a Lie algebra homomorphism$\rho:L\to \Gamma(TM)$into the Lie algebra of vector fields on$M$. The map$\rho$is called an anchor. If the manifold$(M,\{\cdot,\cdot\}_{PB})$is a Poisson manifold, it is natural to require that the vector ... 1 Intuitively, Legendre transform is just "integrate by parts". So, from$pdq - PdQ =F_1$, we have$pdq -d(PQ)+QdP =dF_1$, i.e.,$pdq +QdP =d(F_1+PQ) \equiv dF_2$. The "Transform" means after "integration by parts", we need to change the independent variables. For example, in$F_1$,$p=p(q,Q)$but in$F_2$,$p=p(q,P)$. So we need to solve$Q = Q(q,P)$and ... 1 Isn't it just a convention you have to choose once and for all. It's all about how to pass from Maxwell to lumped element circuits. Especially, I can choose the two different conventions (all quantities are vectorial in the following) $$E=\pm\nabla V$$ since I didn't choose my field-to-potential rules yet. Usually one chooses to conform to the classical ... 1 Consider an element$g$of the symmetry group. Say$g$is represented by a unitary operator on the Hilbertspace $$T_g = \exp(tX)$$ with generator$X$and some parameter$t$. It acts on an operator$\phi(y)$by conjugation $$(g\cdot\phi)(y) = T_g^{-1}\phi(y) T_g = e^{-tX}\phi(y) e^{tX} = \big[ 1 + t[X,\cdot]+\mathcal{O}(t^2)\big]\phi(y)$$ On the other ... 1 I think expecting "fluid" behaviour in terms of a material that does not support shear, is not useful in the context of the various systems you have listed in the question. Instead, I believe you are intuitively connecting ideas and concepts pertaining to conservation laws. So the idea that in specific systems, conserved charges (in the sense of Noether) ... 1 You can start by reading the wikipedia article on the method of characteristics. You will see that in our case the tangent of the characteristic is$(1,-\partial H/\partial q,\partial H/ \partial p)$where the components are in order$t,p,q\$. When you formulate the equation of the characteristic, you will actually find out you get equations of motion of a ...

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