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        Questions tagged [classical-mechanics]

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        integral of the product of an arbitrary function and a linear function [closed]

        In engineering mechanics, a classic approach for the calculation of displacements (virtual work method) requires the evaluation of the definite integral of the product of two continuous functions in $[...
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        203 views

        Global reduction of Hamiltonian with an integral of motion (Poincare' reduction)

        This question is related to a previous one; now I better understand the problem and I can more clearly state what is the question. Background I refer to the following concepts: Liouville ...
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        322 views

        Practical example of Hamiltonian reduction

        I know what is the Liouville integrability: given a Hamiltonian with $n$ degrees of freedom, with $n$ independent constants of motion in involution, the Hamiltonian can be brought to the form $H(p_1, \...
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        66 views

        Deformation gradient conservation law from Lagrangian to Eulerian formulation

        In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...
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        Six yolks in a bowl: Why not optimal circle packing? [closed]

        Making soufflé tonight, I wondered if the six yolks took on the optimal circle packing configuration. They do not. It is only with seven congruent circles that the optimal packing places one in the ...
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        207 views

        Navier-Stokes fluid dynamics, Einstein gravity and holography

        There was some activity a while ago, like 10 years ago, string theoreists try to relate the fluid dynamics, for example, governed by Navier-Stokes equation, to the Einstein gravity, and its ...
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        162 views

        Towards recognizing St. Venant geometrical invariant

        Using partial derivative notation we can express Gauss curvature $K$ in cartesian coordinates: $$\quad p= \partial w/ \partial x, q= \partial w/ \partial y; r=\frac{\partial ^2w}{{\partial {x} ^2} },...
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        Determinant as a Hamiltonian

        Are there two symplectic structures $\omega_1, \omega_2$ on $M_{2n}(\mathbb{R})$ such that the function $Det:M_{2n}(\mathbb{R})\to \mathbb{R}$ is completely integrable with respect to $\omega_{1}$...
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        449 views

        Tying knots via gravity-assisted spaceship trajectories

        Q. Can every knot be realized as the trajectory of a spaceship weaving among a finite number of fixed planets, subject to gravity alone?           To make this more ...
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        1answer
        178 views

        symplectic topology of (perturbed) KAM tori

        Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies ...
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        Is there a convex three-dimensional body with constant width and only finitely-many equilibria? Or: do spheroform gömböcök exist?

        Mathematical questions. The mathematical (and 'gravity'-free) formulation of the question in the title is given by the following questions: Q1. Does there exist $(a,b)\in\omega^2\setminus\{(0,0)\}$ ...
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        164 views

        How to make sense of the Euler Lagrange equations for an infinite action?

        The Euler–Lagrange equation is an equation satisfied by a function $q$, which is a stationary point of the functional $S(\boldsymbol q) = \int_a^b L(t,q(t),\dot{q}(t))\, \mathrm{d}t$ Say we have an ...
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        2answers
        221 views

        Can one obtain this ODE as an Euler-Lagrange equation?

        Some of the second order ODE can be considered as Euler-Lagrange equations for an appropriate Lagrangian. However this is true not for arbitrary second order equation. But some of important equations ...
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        270 views

        Conjecture: Finitely many points where gravitational field due to N masses vanishes

        Given a configuration $C$ of $N$ distinct fixed points of equal mass in the plane (eventually in space), let $f_C(N)$ denote the number of points $P$ for which the gravitational field at $P$ vanishes. ...
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        2answers
        166 views

        A Stochastic Dynamical Billiard

        Consider the following stochastic dynamical system. Fix $a > 0$, $b > 0$ and $v > 0$, and let $\mathbf{r}(t)=(x(t),y(t))$ be the position at time $t$ of a point which moves in the rectangle ...

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