Linked Questions

157
votes
15answers
42k views

What's the point of Hamiltonian mechanics?

I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
70
votes
4answers
21k views

Physical meaning of Legendre transformation

I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
28
votes
4answers
5k 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 ...
11
votes
5answers
2k views

A mathematically illogical argument in the derivation of Hamilton's equation in Goldstein

In the book of Goldstein, at page 337, while deriving the Hamilton's equations (canonical equations), he argues that The canonical momentum was defined in Eq. (2.44) as $p_i = \partial L / \partial \...
12
votes
2answers
1k views

Can the Lagrangian be written as a function of ONLY time?

The lagrangian is always phrased as $L(t,q,\dot{q})$. If you magically knew the equations $q(t)$ and $\dot{q}(t)$, could the Lagrangian ever be written only as a function of time? Take freefall for ...
12
votes
5answers
4k views

Understanding “natural variables” of the thermodynamic potentials using the example of the ideal gas

I'm struggling with the concept of "natural variables" in thermodynamics. Textbooks say that the internal energy is "naturally" expressed as $$ U = U(S,V,N)$$ For an ideal gas, I could take the ...
5
votes
3answers
1k views

Is the shorthand $ \partial_{\mu} $ strictly a partial derivative in field theory?

The Euler-Lagrange equation for particles is given by $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{q}} = \frac{\partial L}{\partial q},\tag{1}$$ and for fields it is $$ \partial_{\mu} \frac{\...
1
vote
2answers
1k views

Independent Variables of a Lagrangian [duplicate]

Let us consider a particle in one spatial dimension $x$ and one temporal dimension $t$. Its Lagrangian $L$ is given by \begin{eqnarray*} L &=& T- V \\ &=& \frac{1}{2} m\dot{x}^2 - ...
2
votes
3answers
585 views

Time translation invariance of Hamiltonian

I am learning about the time translation invariance of the Hamiltonian. I read that the time translation invariance is already manifest in the fact that our Hamiltonian is chosen an ...
5
votes
3answers
538 views

Are generalised coordinates truly independent?

Say we have a system with two generalised coordinates $x$ and $y$. When we solve the equations of motion we find $x=x(t)$ and $y=y(t)$. I can invert one of these solutions to find $t=t(y)$ and ...
1
vote
1answer
352 views

Why does every classical dynamical variable depend on position and momentum rather than position and velocity according to Shankar?

The statement (from Shankar's Quantum Mechanics): Every dynamical variable $\omega(x,p)$ is a function of $p$ and $x$ This means (I think) that a classical system can be totally determined if one ...
3
votes
3answers
158 views

Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?

I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
0
votes
2answers
131 views

Given the action, derive the Lagrangian (Fields and non-fields)

Let us denote $L$ the Lagrangian, and $\mathcal{L}$ the Lagrangian density, and the action $S$. It is common to find the action based on the Lagrangian. Here, however, I am interested in the reverse ...
3
votes
3answers
85 views

Some confusion regarding the specifics of the geometric formulation of Lagrangian mechanics and Noether's theorem

I would like to resolve a few problems I'm having regarding the exact procedure of Lagrangian mechanics when formulated as the tangent bundle of configuration space. These problems are not overly ...
0
votes
1answer
135 views

Is there a quantum analogue of Mean Value Theorem theorem?

Background I was thinking of Mean Value Theorem in the context of classical mechanics I have $2$ points $A$ and $B$ and my particle goes from $A$ to $B$ then I know the velocity of the particle at a ...

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