All Questions
231
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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 ...
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20
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3
answers
881
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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
4
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2
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227
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Are these two expressions for $\mathrm{d}E/\mathrm{d}t$ compatible?
This is purely recreational, but I'm eager to know the answer.
I was playing around with Hamiltonian systems whose Hamiltonian is not equal to their mechanical energy $E$.
If we split the kinetic ...
- 34
2
votes
2
answers
148
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Do dynamic systems that are based on a variational principle imply a conservation law?
In many dynamic systems in classical physics, as well as quantum mechanics, the equation of motion can be derived from a variational principle (VP), i.e. minimizing an action integral of some sort.
I ...
- 261
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48
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Why does nature favour systems that follow from a variational principle? [closed]
When Newton discovered ‘Newton’s law’ he was probably not aware that it could be viewed as a consequence of minimizing an ‘action integral’ (integral of some Lagrangian density).
Since the same is ...
- 261
1
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How is kinetic energy $T$ given by $T=\dfrac{1}{2}\sum_{i}p_{i}\dot{q_{i}}$ in Hamiltonian and Lagrangian mechanics?
Im going through a website teaching Hamiltonian mechanics and I know the below
$$-\dot{p}_{i}=\dfrac{\partial H}{\partial q_{i}} \tag{14.3.12}$$
$$\dot{q}_{i}=\dfrac{\partial H}{\partial p_{i}} \tag{...
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27
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3
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Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?
When we make an arbitrary invertible, differentiable coordinate transformation $$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
2
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Lagrange with Higher Derivatives (Ostrogradsky instability) [duplicate]
In class our teacher told us that, if a Lagrangian contain $\ddot{q_i}$ (i.e., $L(q_i, \dot{q_i}, \ddot{q_i}, t)$) the energy will be unbounded from below and it can take any lower values (in other ...
- 3,122
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How to solve for the velocities when calculating the conjugate momenta in special relativity?
I try to get the momenta $$p_{\sigma} = \frac{\partial L}{\partial \dot{x}^{\sigma}}$$ from the free one particle Lagrangian $$L = -mc\sqrt{-\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ I got to the ...
- 147
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225
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What is the physical significance of $ -\frac{\partial L}{\partial t} = \frac{\partial H}{\partial t} $?
Let's assume a conservative holonomic system with $n$ independent generalized coordinates and a Lagrangian $L(q,\dot{q},t)$, resulting in a system of $n$ Lagrange differential equations of second ...
- 209
2
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1
answer
86
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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
3
votes
3
answers
657
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Hamiltonian Mechanics without a Lagrangian
Let's say I want to develop Hamiltonian mechanics from scratch without going through Lagrangian mechanics and Legendre transformations. How would I go about doing that? What I am struggling with is a ...
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74
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Lagrangian with Higher Order derivatives and Ignorable coordinate
I have a Lagrangian for a physical system that ends up being function of one generalized coordinate $q_1$, its generalized velocity $\dot{q}_1$ and its generalized (?) accelration $\ddot{q}_1$, and ...
2
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3
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285
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Can someone explain conservation laws in terms of state space?
"Whenever a dynamical law divides the state space into separate cycles, there is a memory of which cycle they started in. Such a memory is called a conservation law."
—What is meant by this ...
1
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2
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233
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Hamiltonian formalism in Goldstein's matrix representation, chap. 8.1
There are several points over which I stumble when studying Goldstein, 3rd ed., chap 8.1, concerning the matrix representation of the hamiltonian formalism. In (8.22) he assumes the lagrangian to be (...
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