All Questions
26
questions
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
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
10
votes
3
answers
4k
views
Is there a valid Lagrangian formulation for all classical systems?
Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths?
On the wikipedia page of Lagrangian mechanics, there is an ...
- 8,523
8
votes
5
answers
716
views
Why can't we obtain a Hamiltonian from the Lagrangian by only substituting?
This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't ...
- 590
8
votes
1
answer
2k
views
If the Lagrangian depends explicitly on time then the Hamiltonian is not conserved?
Why is the Hamiltonian not conserved when the Lagrangian has an explicit time dependence? What I mean is that it is very obvious to argue that if the Lagrangian has no an explicit time dependence $L=L(...
- 227
7
votes
3
answers
324
views
Hamiltonian of non-regular Lagrangian is well-defined on phase space
In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian
$$H=\dot{q}^np_n-L,\tag{1.8}$$
although trivially a function of $q$ and $\dot{q}$, can ...
- 3,915
6
votes
1
answer
924
views
Independence of generalised coordinates and momenta in Hamiltonian mechanics [duplicate]
I am told that in Hamiltonian mechanics, we put the generalised coordinates $q_i$ and generalised momenta $p_i$ on equal footing, and treat them as being independent from one another. But I'm ...
- 379
4
votes
1
answer
806
views
Do time-invariant Hamiltonians define closed systems?
In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system?
Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
- 155
4
votes
2
answers
801
views
Liouville's theorem for systems with dissipation described by a single hamiltonian
Following this link, one can treat dissipation by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation:
$$
L[q, \dot{q}, t] = e^{\frac{t \beta}{ m}...
- 6,763
3
votes
3
answers
704
views
Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?
I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
- 11.8k
3
votes
1
answer
1k
views
General Form for Kinetic Energy Given Velocity Independent Potential such that $\mathcal{H}=E$
Suppose the potential energy is independent of $\dot{q},$ i.e $\frac{\partial V}{\partial\dot{q}}=0$. What is the most general form of the kinetic energy such that the Hamiltonian is the total energy? ...
- 32
3
votes
0
answers
222
views
Does the additivity property of Integrals of motion and Lagrangians valid in all situations?
I would like to know if the additivity property of an integral (constant) of motion valid in all situations ? It works for energy but does it work for all other integrals of motion in all kinds of ...
- 345
2
votes
1
answer
861
views
Proof that Hamiltonian is constant if Lagrangian doesn't depend explicitly on time
I know that on solutions of motion we have $\frac{dH}{dt}=\frac{\partial H}{\partial t} $ and i understand the proof for this fact. Then, we have that $$\frac{\partial H}{\partial t}=-\frac{\partial L}...
- 49
2
votes
2
answers
156
views
How to derive the fact that $p\sim d/dx$ and $H\sim d/dt$ from classical mechanics?
I am trying to understand Noether's conserved quantities to shifts in time and or position. I have seen the derivation of the operators for Schrodinger's equation but not for classical mechanics.
Is ...
- 381