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### Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Lagrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian ...
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### Snell's law in vector form

Snell's law of refraction at the interface between 2 isotropic media is given by the equation: \begin{equation} n_1 \,\text{sin} \,\theta_1 = n_2 \, \text{sin}\,\theta_2 \end{equation} where $\theta_1$...
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### Why do we need the supremum when performing Legendre transformations?

Legendre transforms appear all over physics. For instance, in statistical mechanics, they allow us to move between descriptions in terms of different thermodynamic potentials. Similarly, in quantum ...
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### Why are Hamiltonian Mechanics well-defined?

I have encountered a problem while re-reading the formalism of Hamiltonian mechanics, and it lies in a very simple remark. Indeed, if I am not mistaken, when we want to do mechanics using the ...
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### Besides the 2nd law of thermodynamics, what laws of optics prevent the temperature of the focal point of lens from being hotter than the light source?

I'm pretty sure that you can't take a magnifying glass and make it focus to a point that is hotter than the surface of your light source. For example, when you're outside trying to fry ants with your ...
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### Lagrangian and negative mass

How can one see that a Lagrangian in case of a free particle dictates that the mass can't be negative? Consider for example the case where the Lagrangian is given by $${\cal L}=\frac{1}{2} mv^2 .$$