Linked Questions

29
votes
4answers
6k views

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 ...
14
votes
2answers
10k views

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 $\theta_1$...
8
votes
2answers
744 views

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 ...
17
votes
1answer
843 views

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 ...
11
votes
1answer
1k views

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 ...
6
votes
2answers
1k views

Legendre transformation: non-convex/non-convave functions

The Legendre transformation is used to derive the Hamiltonian from the Lagrangian, and it finds many applications in thermodynamics to convert between the different potentials. $ f(x) \rightarrow g(u)...
6
votes
1answer
934 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
9
votes
2answers
439 views

Is symplectic form in Hamiltonian mechanics a physical quantity?

Is symplectic form $dp_i \wedge dq_i$ in Hamiltonian mechanics a physical quantity? It feels to me to be something different than say energy, momentum or mass. Like just certain structure. The real ...
6
votes
1answer
298 views

What are the restrictions on the Hamiltonian in QM?

In quantum mechanics, we usually write the Hamiltonian as: $$\hat{H}=\hat{T}+\hat{V}$$ But in classical mechanics, there are several reasons why it would not have this form: We've chosen some ...
3
votes
1answer
278 views

Proof of Validity of Thick Lens Model by Hamiltonian Formalism or Otherwise

The paraxial imaging geometry of any axially symmetric optical system can be described by three real parameters: the axial positions of the two principal planes and one focal length (i.e. $f_f=f_b$ in ...
2
votes
1answer
499 views

Why is the formula for stationary action expressed as kinetic minus potential energy instead of potential minus kinetic energy?

I am sure this is a duplicate but I could not spot it exactly. And I am sure folks have covered this topic online here in great detail. I am referring to the Lagrangian here in the "Action" formula ...
4
votes
2answers
117 views

Changing sign of Lagrangian & Hamiltonian: how to interpret energies then?

In Lagrangian mechanics, it is possible to multiply the Lagrangian by a constant $a$. Let's assume I take $a=-1$. Then, the Hamiltonian will have its sign changed as well. And it will represent the ...
0
votes
0answers
170 views

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 $$ {\cal L}=\frac{1}{2} mv^2 . $$
6
votes
0answers
126 views

Lagrangians, Hamiltonians and the Legendre Transform [duplicate]

I have been studying the Lagrangian formulation of classical mechanics. I have been introduced to the Hamiltonian formulation via the Legendre transform, and studied the transform from this excellent ...
2
votes
0answers
129 views

Lagrangian and convexity

Is it possible to model any physical system with a Lagrangian convex in its velocity variables ? I am aware that many Lagrangian can model the same system and maybe not all of them are partially ...

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