All Questions
231
questions
-1
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0
answers
34
views
What is the relationship between phase space and Jacobian in Nakahara Eq.1.15 (under this equation)?
In Nakahara's Geometry, topology and physics, under Eq.1.15 they give an equation
\begin{align*}
\det\left(\frac{\partial p_i}{\partial\dot{q}_j}\right)=\det\left(\frac{\partial^2L}{\partial\dot{q}_i\...
- 37
-3
votes
2
answers
81
views
Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
In Lagrangian mechanics the momentum is defined as:
$$p=\frac{\partial \mathcal{L}}{\partial \dot q}$$
Also we can define it as:
$$p=\frac{\partial S}{\partial q}$$
where $S$ is Hamilton's principal ...
- 443
0
votes
1
answer
97
views
Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
I am solving problem 19 of ch 10 of Goldstein mechanics. The problem is:
A three-dimensional harmonic oscillator has the force constant k1 in the x- and y- directions and k3 in the z-direction. Using ...
- 13
1
vote
1
answer
56
views
Confusing Goldstein Statement about Magnitude of the Lagrangian
On page 345 of Goldstein's Classical Mechanics 3rd Ed., he writes:
...the Hamiltonian is dependent both in magnitude and in functional form upon the initial choice of generalized coordinates. For the ...
- 7,398
2
votes
0
answers
73
views
Why can't we treat the Lagrangian as a function of the generalized positions and momenta and vary that? [duplicate]
Some background: In Lagrangian mechanics, to obtain the EL equations, one varies the action (I will be dropping the time dependence since I don't think it's relevant) $$S[q^i(t)] = \int dt \, L(q^i, \...
- 93
2
votes
1
answer
81
views
What is the benefit of Hamiltonian formalism to promote ($q,\dot{q},t$) from Lagrangian to ($q,p,t$) despite getting the same EOM finally? [duplicate]
Hamiltonian formalism follows
$$H(q,p,t)=\sum_i\dot{q_i}p_i-L(q_i,\dot{q}_i,t) $$ and $$\dot{p}=-\frac{\partial H}{\partial q}, \dot{q}=\frac{\partial H}{\partial p} $$
but finally these will get the ...
- 311
0
votes
1
answer
69
views
Changing variables from $\dot{q}$ to $p$ in Lagrangian instead of Legendre Transformation
This question is motivated by a perceived incompleteness in the responses to this question, which asks why we can't just substitute $\dot{q}(p)$ into $L(q,\dot{q})$ to convert it to $L(q,p)$, which ...
- 7,398
1
vote
1
answer
33
views
Regarding varying the coefficients of constraints in the pre-extended Hamiltonian in Dirac method
consider the following variational principle:
when we vary $p$ and $q$ independently to find the equations of motion, why aren't we explicitly varying the Coeff $u$ which are clearly functions of $p$ ...
1
vote
0
answers
49
views
Poincare-Cartan form of charged particle in electromagnetic field
In the paper by Littlejohn, 1983, the canonical Hamiltonian $h_c$ of a charged particle in electromagnetic field is given by,
$$
h_c (\vec{q}, \vec{p}, t) = \frac{1}{2m} \left[ \vec{p} - \frac{e}{c} \...
- 494
5
votes
1
answer
592
views
Does the Hamiltonian formalism yield more Noether charges than the Lagrangian formalism?
In Lagrangian formalism, we consider point transformations $Q_i=Q_i(q,t)$ because the Euler-Lagrange equation is covariant only under these transformations. Point transformations do not explicitly ...
- 145
0
votes
1
answer
80
views
Lagrangian and Hamiltonian Mechanics: Conjugate Momentum
I am a physics undergraduate student currently taking a classical mechanics course, and I am not able to understand what conjugate/canonical momentum is (physically). It is sometimes equal to the ...
2
votes
3
answers
148
views
Derivation of Hamiltonian by constraining $L(q, v, t)$ with $v = \dot{q}$
I am trying to reconstruct a derivation that I encountered a while ago somewhere on the internet, in order to build some intuition both for $H$ and $L$ in classical mechanics, and for the operation of ...
- 23
1
vote
1
answer
85
views
Hamiltonian analysis of relational $N$-Particle Dynamics
I am following "A Shape Dynamics Tutorial, Flavio Mercati" (https://arxiv.org/abs/1409.0105), and have problems understanding the hamiltonian formulation of $N$-particle dynamics as sketched ...
- 513
3
votes
3
answers
788
views
Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics
In Lagrangian mechanics we have the Euler-Lagrange equations, which are defined as
$$\frac{d}{dt}\Bigg(\frac{\partial L}{\partial \dot{q}_j}\Bigg) - \frac{\partial L}{\partial q_j} = 0,\quad j = 1, \...
- 3,350
0
votes
1
answer
48
views
How to determine which coordinates to use for calculating the Hamiltonian? [closed]
In my classical mechanics course, I was tasked with finding the Hamiltonian of a pendulum of variable length $l$, where $\frac{dl}{dt} = -\alpha$ ($\alpha$ is a constant, so $l = c - \alpha t$.).
I ...
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