All Questions
231
questions
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
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
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
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 ...
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 ...
27
votes
3
answers
3k
views
Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?
When we make an arbitrary invertible, differentiable coordinate transformation $$s_i=s_i(q_1,q_2,...q_n,t),\forall i,$$ the Lagrange's equation in terms of old coordinates $$\frac{d}{dt}\left(\frac{\...
- 11.8k
27
votes
2
answers
23k
views
What is the difference between configuration space and phase space?
What is the difference between configuration space and phase space?
In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's ...
- 2,324
25
votes
3
answers
29k
views
Constructing Lagrangian from the Hamiltonian
Given the Lagrangian $L$ for a system, we can construct the Hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
- 26.8k
25
votes
1
answer
3k
views
What's the physical intuition for symplectic structures?
I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector field....
- 1,043
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
21
votes
2
answers
2k
views
Apparent paradox between Lagrangian and Hamiltonian formulations of classical mechanics
I've recently come across a strange result when comparing the Hamiltonian and Lagrangian formulations of classical mechanics.
Suppose we are working in the regime where we can say the Hamiltonian $H$ ...
- 351
20
votes
2
answers
825
views
Are there other less famous yet accepted formalisms of Classical Mechanics? [duplicate]
I was lately studying about the Lagrange and Hamiltonian Mechanics. This gave me a perspective of looking at classical mechanics different from that of Newton's. I would like to know if there are ...
- 3,905
20
votes
3
answers
881
views
What properties make the Legendre transform so useful in physics?
The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e....
- 14.8k
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