Questions tagged [classical-mechanics]
For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question.
2,185 questions
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Are the motion equations of an optimal control problem geodesics on a manifold?
Let us consider a Lagrangian system for which the equations of motion come from Hamilton's principle and are such that the control variable $\tau$ equals the equations of motion of a free system, ...
- 534
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Strain Tensor Based on Clifford Algebra
Let $u=u_i e_i$ be the displacement field of a continuum body. Then the displacement gradient tensor H based on classical formulation is given by $H=\nabla u = u_{i,j}\, e_i \otimes e_j$, where $\...
3
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1
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Probability of rolling a specified face for Archimedean solids
Here it is shown that (for a "suitable" mathematical definition of fairness) there are no fair $n$-sides die with odd $n$. The question originated with fairness being defined as, among other,...
- 743
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Which orientation of a truncated cone will drain faster? [migrated]
A container shaped like a truncated cone (frustum) as in cylinder-like shape where one end has a larger radius than the other. Both orientations contain the same volume of water, and the outlet hole (...
- 3
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0
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How do concentrated (nodal) forces manifest in the weak form of the linear elasticity pde?
Consider a classical BVP governed by linear elasticity
$$
\begin{align*}
-\nabla \cdot \boldsymbol{\sigma} = \boldsymbol{b} & \quad \textrm{in} \nobreakspace \nobreakspace \Omega \subset \...
- 11
3
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0
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68
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Coupled Oscillators or "Springs and Masses" (question about sum of binomial coefficients and a reference request)
Suppose $n$ identical masses (each with mass $m$) lie on a friction-less surface and are connected by springs (each with spring constant $k$ and equilibrium length $a$). Let the positions of the ...
- 355
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Can it be proven that CVT(Continuously Variable Transmissions) are impossible using only smooth rigid bodies?
In mechanical engineering, there are many designs of a CVT(Continuously Variable Transmissions) that is able to change gear ratio continuously, but all of them are unsatisfactory for some reason.
...
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Geodesics on spheres (from Taylor’s classical mechanics) [duplicate]
I'm tying to find where I went Wrong in my efforts to solve problem 6.1 in Taylor’s book on classical mechanics.
Using spherical polar coordinates $(r,\theta,\phi)$, show that the length of a path ...
- 363
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1
answer
67
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Klein Gordon equation using the Hamiltonian Formalism
Background.
I am working through "Mechanics and Symmetry" by Marsden and Ratiu. In Ch. 3.2, they derive the Klein-Gordon equation using a Hamiltonian of the form
$$H(\varphi,\pi)=\int_{\...
- 137
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0
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When is a polynomial in |A| polyconvex?
When is $W(A) = p(|A|)$ a polyconvex function, where $p$ is a polynomial and |A| is the Frobenius norm of the matrix $A$?
A sufficient condition: because $|A|$ is convex in the entries of $A$, then $W$...
- 38
1
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1
answer
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Sufficient properties of vector addition?
How can we prove the following theorem?
Theorem: If $\oplus : R^3 \times R^3 \to R^3$ satisfies:
(i) $u \parallel v \implies u \oplus v = u+v$
(ii) $u \oplus v = v \oplus u$
(iii) $u \oplus (v \oplus ...
- 355
2
votes
1
answer
201
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Computing the first variation of a variational integral
Having started to read Giaquinta/Hildebrandt "Calculus of Variations I", 2 ed. 2004, in order to get a better mathematical foundation for my understanding of the Hamilton principle of least ...
- 545
2
votes
1
answer
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Period of Periodic Solutions to Hamiltonian System in Plane
Consider a planar Hamiltonian system with Hamiltonian $H(x,y)$, i.e. a system given by $$x^\prime (t) = \frac{\partial H}{\partial y}$$ $$y^\prime(t) = - \frac{\partial H}{\partial x}. $$ Suppose $\...
- 1,158
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How can I derive angular velocity components using Euler angle parameters & purely using the Euler rotational matrices?
I'm using Goldstein's Classical Mechanics 3rd Ed. In Section 4.9 - Rate of Change of a Vector he derives the components of the angular velocity vector for the rate of vector change transformation ...
- 1
2
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1
answer
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Deriving a quadratic function from a table of values
The number of revolutions per second (rps) of a flywheel slowing down is given in terms of time in the following table:
Time (s)
Revolutions/second
0
360
30
351
60
324
90
279
120
216
150
135
Find the ...
- 385