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3

If we start with the equation you quote: $$\rho_c = \frac{3H^2}{8\pi G}$$ and rewrite it as: $$\rho_c = \frac{3}{8\pi G} H^2$$ then it's the same as your equation: $$\rho_c = E H^2$$ because all you've done is to replace the constant factor of $3/8\pi G$ with the symbol $E$. This is of couse a perfectly reasonable thing to do, but it isn't new in ...

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The hamiltonian is a function on the phase space so it transforms as such. I.e. $$K(y(x))=H(x)$$ You can derive the condition on the transformation of J precisely from the condition $$x'(t) = J \nabla_x H(x(t)) \to y'(t) = J \nabla_{y(x)} K(y(x))$$

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Too long for a comment, so in an answer: Not all the springs are a function of $x_1$ and $x_2$, only spring $K_b$ is a function of both $x_1$ and $x_2$. Spring force is a function of how much a spring is stretched, e.g. how much difference er is between the beginning and the end of a spring, so $x_{begin}$-$x_{end}$ or $x_{top}$-$x_{bottom}$ for this case. ...

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It's just a matter of conservation of volume. You need to start by stating assumptions that are obvious, prima-facia, that anyone would agree with, such as the liquid being incompressible. Then you say, if one accepts these assumptions, then do like @User58220 said, assume the contrary and derive a contradiction. Suppose you have to explain it to someone, ...

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If motor A does the same job as motor B, but with a 10x greater load, and the mechanical advantage (gearing etc) is the same, then I would expect that the torque that A supplies is ten times greater as well. But that is not quite how you phrased the question. It necessarily follows that a higher HP motor can supply greater torque - at least, with the right ...

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An (undriven) damped harmonic oscillator (satisfying $m\ddot{x} + \gamma \dot{x} + \omega_0^2 x=0$) can be solved by the solution(s) $x_0e^{i \omega t}$. For an underdamped oscillator these solutions represent pure oscillations mixed with exponential decay(/growth). Because both solutions for $\omega$ oscillate with the same period, all combinations of them ...

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Classical physics is developed based on this idea that understanding of a phenomenon will help in predicting the future and the past of a system. It actually works on the prejudice that quantities which are measurable can be measured with utmost accuracy by developing better methodology for experiments. So, the based on classical physics i.e.ignoring the ...

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In classical mechanics, specifying the initial coordinates and velocities of all particles uniquely determines the system's future ... Uniqueness and existence are not guaranteed in classical mechanics. Physicists gloss over the cases where uniqueness is not guaranteed. Initial positions and velocities suffice if the resultant accelerations are ...

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Assume the contrary, the volume of liquid displaced is greater than, or less than, the volume of the object. Then there is either a volume containing neither water nor object, or there is a volume occupied by both water and object.

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Damping an oscillation changes the frequency in two ways. The oscillation can no longer be said to take place at a single frequency, but covers a continuous distribution of frequencies characterised by a Lorentzian profile, with a width that increases with the damping. Second, the peak of this distribution occurs at a lower frequency than the natural ...

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