All Questions
64
questions
35
votes
2
answers
10k
views
Lagrangian and Hamiltonian EOM with dissipative force
I am trying to write the Lagrangian and Hamiltonian for the forced Harmonic oscillator before quantizing it to get to the quantum picture. For EOM $$m\ddot{q}+\beta\dot{q}+kq=f(t),$$ I write the ...
user56199
18
votes
2
answers
5k
views
Constraints of relativistic point particle in Hamiltonian mechanics
I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint:
$$S=-m\...
- 1,565
203
votes
15
answers
57k
views
What's the point of Hamiltonian mechanics?
I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
- 28.3k
96
votes
4
answers
32k
views
Physical meaning of Legendre transformation
I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
- 1,227
29
votes
9
answers
25k
views
Book about classical mechanics
I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
22
votes
3
answers
3k
views
Why exactly do we say $L = L(q, \dot{q})$ and $H = H(q, p)$?
In classical mechanics, we perform a Legendre transform to switch from $L(q, \dot{q})$ to $H(q, p)$. This has always been confusing to me, because we can always write $L$ in terms of $q$ and $p$ by ...
- 103k
38
votes
3
answers
6k
views
Are the Hamiltonian and Lagrangian always convex functions?
The Hamiltonian and Lagrangian are related by a Legendre transform:
$$
H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t).
$$
For this to be a Legendre ...
- 34.3k
16
votes
3
answers
6k
views
Hamilton-Jacobi Equation
In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
- 921
12
votes
5
answers
3k
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 \...
- 2,283
34
votes
4
answers
28k
views
Any good resources for Lagrangian and Hamiltonian Dynamics?
I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics.
So far at my university ...
19
votes
1
answer
1k
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 ...
- 2,296
10
votes
1
answer
2k
views
How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?
Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
- 411
4
votes
4
answers
490
views
Action principle, Lagrangian mechanics, Hamiltonian mechanics, and conservation laws when assuming Aristotelian mechanics $F=mv$
Define a physical system when Aristotelian mechanics $F=mv$ instead of Newtonian mechanics $F=ma$.
Then we could have action $I=\int L(q,t)dx$ rather than $\int L(q',q,t)dx$.
Is there an action ...
- 417
12
votes
2
answers
878
views
Why do we use the Lagrangian and Hamiltonian instead of other related functions?
There are 4 main functions in mechanics. $L(q,\dot{q},t)$, $H(p,q,t)$, $K(\dot{p},\dot{q},t)$, $G(p,\dot{p},t)$. First two are Lagrangian and Hamiltonian. Second two are some kind of analogical to ...
- 353
9
votes
2
answers
3k
views
Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian
The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics.
Adding the total time derivative of a function of $q_i$ and t to the Lagrangian ...
- 181