All Questions
26
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
64
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
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 $$\frac{\partial H}{\partial t}=-\frac{\partial L}...
- 49
0
votes
2
answers
42
views
Is there a way to understand which variable is more influential in the dynamics of a system?
Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two ...
2
votes
2
answers
156
views
How to derive the fact that $p\sim d/dx$ and $H\sim d/dt$ from classical mechanics?
I am trying to understand Noether's conserved quantities to shifts in time and or position. I have seen the derivation of the operators for Schrodinger's equation but not for classical mechanics.
Is ...
- 381
0
votes
1
answer
240
views
A relationship between Lagrangian formalism and Hamiltonian formalism
In the Lagrangian formalism, The Lagrangian
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
The equations of motion for a given system is given by minimizing the action functional which ...
- 11
20
votes
3
answers
881
views
What properties make the Legendre transform so useful in physics?
The Legendre transform plays a pivotal role in physics in its connecting Lagrangian and Hamiltonian formalisms. This is well-known and has been discussed at length in this site (related threads are e....
- 14.8k
2
votes
1
answer
86
views
I don't get this "derivation" of canonical transformation
Given a transformation
$$(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).$$
Now, what I've ...
- 1,951
0
votes
1
answer
237
views
Why is the energy function not always equal to total energy? [duplicate]
Why is the energy function $h = \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $ not always equal to total energy $E = T + V$? Here $T$ is Kinetic Energy and $V$ is Potential Energy. I've read ...
- 75
3
votes
3
answers
704
views
Why is the Legendre transformation the correct way to change variables from $(q,\dot{q},t)\to (q,p,t)$?
I always found Legendre transformation kind of mysterious. Given a Lagrangian $L(q,\dot{q},t)$, we can define a new function, the Hamiltonian, $$H(q,p,t)=p\dot{q}(p)-L(q,\dot{q}(q,p,t),t)$$ where $p=\...
- 11.8k
7
votes
3
answers
324
views
Hamiltonian of non-regular Lagrangian is well-defined on phase space
In section 1.1.3 of Quantization of Gauge Systems by Henneaux and Teitelboim, it is stated that the Hamiltonian
$$H=\dot{q}^np_n-L,\tag{1.8}$$
although trivially a function of $q$ and $\dot{q}$, can ...
- 3,915
0
votes
0
answers
69
views
When the hamiltonian isn't equal to energy? [duplicate]
I have the following hamiltonian:
$$H = \frac{p_1^2}{2}+\frac{(p_2-k\;q_2)^2}{2} ,\qquad k\in\mathbb{R}.$$
I know that the hamiltonian isnt explicitly dependent on time so $H$ is a motion ...
1
vote
1
answer
139
views
What if we set Hamilton-Jacobi equation as an axiom?
We usually postulate the principle of least action. Then we can get Lagrangian mechanics. After that we can get Hamiltonian mechanics either from postulate or from the equivalent Lagrangian mechanics. ...
- 353
1
vote
0
answers
30
views
How to calculate Hamiltonian when Lagrangian has higher order derivatives? [duplicate]
If we have a Lagrangian density $\mathcal{L}$ for a scalar field $\phi$ depending on $\phi$, $\partial _{\mu} \phi$, and $\partial _{\mu} \partial _{\nu} \phi$, what is the Hamiltonian? Additionally, ...
- 591
2
votes
1
answer
1k
views
Difference between the energy and the Hamiltonian in a specific example
The problem is the following:
Consider a particle of mass $m$ confined in a long and thin hollow pipe, which rotates in the $xy$ plane with constant angular velocity $\omega$. The rotation axis ...
- 1,045
2
votes
4
answers
1k
views
The definition of the hamiltonian in lagrangian mechanics
So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
user63248
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
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
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
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
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
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
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
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 ...
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