All Questions
25
questions
1
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0
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83
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Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
- 133
2
votes
2
answers
153
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What is the most general transformation between Lagrangians which give the same equation of motion?
This question is made up from 5 (including the main titular question) very closely related questions, so I didn't bother to ask them as different/followup questions one after another. On trying to ...
- 785
4
votes
2
answers
137
views
Gauge Symmetry of the Lagrangian
My teacher told the following statement to me during office hours. Is it correct and if so, how could one go about proving it?
Given a material system subject to holonomic and smooth constraints ...
- 402
13
votes
2
answers
407
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Anticommutation of variation $\delta$ and differential $d$
In Quantum Fields and Strings: A Course for Mathematicians, it is said that variation $\delta$ and differential $d$ anticommute (this is only classical mechanics), which is very strange to me. This is ...
3
votes
3
answers
702
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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
0
votes
0
answers
409
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Invertibility of the Legendre Transformation
The above image shows the Legendre Transformation in the context of an introduction to the Hamiltonian formalism.
My question is in 4.6, wherein $u(x, y)$ has been defined; what is the guarantee ...
- 317
2
votes
1
answer
142
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Arnold's holonomic constraints being limits of potential energy
The following quote comes from Arnold's "Mathematical methods in mechanics" book:
"We consider potential energy $U_N = Nq_2^2 + U_0(q_1, q_2) $, depending
on parameter $N$ (which we ...
- 399
1
vote
1
answer
128
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Meaning and Origin of an Expression which Involves Virtual Displacement
As an additional point of confusion related to the answer given here:
Confusion with Virtual Displacement
I have encountered the following expression in my study of virtual displacements.
$$\delta{...
- 348
5
votes
2
answers
2k
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Confusion with Virtual Displacement
I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
- 348
3
votes
1
answer
992
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Is the phase space a linear vector space?
Is the phase space in classical mechanics a linear vector space (LVS)? If yes,, can we define operators, inner products in this space?
Edit: I have seen Liouville operator $\mathcal{L}$ in Classical ...
- 26.7k
1
vote
2
answers
86
views
Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]
Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
- 1,692
3
votes
0
answers
54
views
Action for solution of general nth order differential equation [duplicate]
Suppose I want to find solution to a general nth order differential equation.
(If I am right about the logic then) one might say that the solution $y\equiv y(x)$ is that function for which the ...
- 957
11
votes
2
answers
1k
views
Is the Legendre transformation a unique choice in analytical mechanics?
Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
- 1,281
25
votes
3
answers
29k
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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 ...
- 26.7k
4
votes
1
answer
381
views
How does one express a Lagrangian and Action in the language of forms?
In Lipschitzs Classical Mechanics a Lagrangian is defined as:
$L(q,q',t)$ for some trajectory $q(t)$ of a particle
And the action is defined as:
$S:=\int^a_b L(q,q',t) dt$
How does one express ...
- 13.1k