All Questions
27
questions
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
1
vote
0
answers
83
views
Independence of variables in Lagrangain and Hamiltonian mechanics - Rigorous Mathematical approach
I am trying to self-learn the Hamiltonian and Lagrangian mechanics and I came across thoughts to which I could not find an answer therefore I would like to try and ask them here.
My questions are as ...
- 23
0
votes
0
answers
63
views
What would make the Legendre transformation interesting, from the graphical point of view?
The term "transformation" is often used in physics and mathematics for functions to denote a different (and useful) way of encoding the information in a function. In this question, I want to ...
- 2,599
0
votes
0
answers
38
views
The right domain for Hamiltonians
This question came to me today, and I am now intrigued about it. For a system described by a Lagrangian $L$, the associated Hamiltonian is its Legendre transform. Suppose we consider a given ...
- 585
2
votes
1
answer
852
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 $$\frac{\partial H}{\partial t}=-\frac{\partial L}...
- 49
0
votes
2
answers
42
views
Is there a way to understand which variable is more influential in the dynamics of a system?
Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two ...
2
votes
2
answers
156
views
How to derive the fact that $p\sim d/dx$ and $H\sim d/dt$ from classical mechanics?
I am trying to understand Noether's conserved quantities to shifts in time and or position. I have seen the derivation of the operators for Schrodinger's equation but not for classical mechanics.
Is ...
- 381
0
votes
1
answer
238
views
A relationship between Lagrangian formalism and Hamiltonian formalism
In the Lagrangian formalism, The Lagrangian
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
The equations of motion for a given system is given by minimizing the action functional which ...
- 11
20
votes
3
answers
874
views
What properties make the Legendre transform so useful in physics?
The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e....
- 14.7k
2
votes
1
answer
85
views
I don't get this "derivation" of canonical transformation
Given a transformation
$$(q, p, t)\to (Q(q, p, t), P(q, p, t), t),$$
the modified Hamiltonian, $K$ is related to the original one, $H$, as
$$H(q, p, t) = K(Q(q, p, t), P(q, p, t), t).$$
Now, what I've ...
- 1,951
0
votes
1
answer
235
views
Why is the energy function not always equal to total energy? [duplicate]
Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
- 75
3
votes
3
answers
702
views
Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?
I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
- 11.8k
7
votes
3
answers
324
views
Hamiltonian of non-regular Lagrangian is well-defined on phase space
In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian
$$H=\dot{q}^np_n-L,\tag{1.8}$$
although trivially a function of $q$ and $\dot{q}$, can ...
- 3,895
0
votes
0
answers
69
views
When the hamiltonian isn't equal to energy? [duplicate]
I have the following hamiltonian:
$$H = \frac{p_1^2}{2}+\frac{(p_2-k\;q_2)^2}{2} ,\qquad k\in\mathbb{R}.$$
I know that the hamiltonian isnt explicitly dependent on time so $H$ is a motion ...
1
vote
1
answer
139
views
What if we set Hamilton-Jacobi equation as an axiom?
We usually postulate the principle of least action. Then we can get Lagrangian mechanics. After that we can get Hamiltonian mechanics either from postulate or from the equivalent Lagrangian mechanics. ...
- 353