All Questions
Tagged with classical-mechanics lagrangian-formalism
1,464
questions
203
votes
15
answers
57k
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What's the point of Hamiltonian mechanics?
I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
- 28.3k
147
votes
8
answers
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Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
- 2,145
128
votes
10
answers
41k
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Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
- 8,520
96
votes
4
answers
32k
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Physical meaning of Legendre transformation
I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
- 1,227
76
votes
7
answers
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What is the difference between Newtonian and Lagrangian mechanics in a nutshell?
What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
- 941
67
votes
5
answers
8k
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Is there a Lagrangian formulation of statistical mechanics?
In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and ...
- 34.3k
57
votes
7
answers
9k
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Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,826
49
votes
8
answers
15k
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Classical mechanics without coordinates book
I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
48
votes
5
answers
4k
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Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the Euler-...
- 1,340
41
votes
7
answers
11k
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Is there a proof from the first principle that the Lagrangian $L = T - V$?
Is there a proof from the first principle that for the Lagrangian $L$,
$$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$
in classical mechanics? Assume that Cartesian coordinates are used. ...
- 751
38
votes
3
answers
6k
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Are the Hamiltonian and Lagrangian always convex functions?
The Hamiltonian and Lagrangian are related by a Legendre transform:
$$
H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t).
$$
For this to be a Legendre ...
- 34.3k
37
votes
6
answers
66k
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What are holonomic and non-holonomic constraints?
I was reading Herbert Goldstein's Classical Mechanics. Its first chapter explains holonomic and non-holonomic constraints, but I still don’t understand the underlying concept. Can anyone explain it to ...
- 381
36
votes
3
answers
25k
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Deriving the Lagrangian for a free particle
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.
Proving that a free ...
- 463
36
votes
4
answers
21k
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What exactly is a virtual displacement in classical mechanics?
I'm reading Goldstein's Classical Mechanics and he says the following:
A virtual (infinitesimal) displacement of a system refers to a change in the configuration of the system as the result of any ...
- 36.4k
34
votes
4
answers
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Any good resources for Lagrangian and Hamiltonian Dynamics?
I'm taking a course on Lagrangian and Hamiltonian Dynamics, and I would like to find a good book/resource with lots of practice questions and answers on either or both topics.
So far at my university ...