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Questions tagged [variational-principle]

Any of several principles that find the physical trajectory of a system by minimizing or maximizing some value computed over the proposed path (for instance geometric optics can be reproduced by insisting on a minimum time principle).

See also and .

1213 questions
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Why are there only derivatives to the first order in the Lagrangian?

Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded? Is there a good reason for this or is it simply "because it works".
Sam
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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 them as independent variables?
grizzly adam
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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 functional, require that it be a minimum (or maximum), and…
Jonathan Gleason
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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 momenta for a potentially very large number of…
N. Virgo
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64
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8 answers

Why should an action integral be stationary? On what basis did Hamilton state this principle?

Hamilton's principle states that a dynamic system always follows a path such that its action integral is stationary (that is, maximum or minimum). Why should the action integral be stationary? On what basis did Hamilton state this principle?
tsudot
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53
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5 answers

Derivation of Maxwell's equations from field tensor lagrangian

I've started reading Peskin and Schroeder on my own time, and I'm a bit confused about how to obtain Maxwell's equations from the (source-free) lagrangian density $L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$ (where $F^{\mu\nu} = \partial^\mu A^\nu -…
amc
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5 answers

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-Lagrange equations from the principle of least…
Deep Blue
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How does Fermat's principle make light choose a straight path over a short path?

This is a thought experiment where I have made a "C" shaped hole inside diamond. The refractive index $(\mu)$ of diamond is 2.45. Say we shine a laser from top of the "C" as shown. My calculations show that light reaching A can reach B in the least…
Rishab Navaneet
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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. Among the combinations, $L = T - nV$, only $n=1$…
Chin Yeh
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4 answers

Entropy and the principle of least action

Is there any link between the law of maximum entropy and the principle of least action. Is it possible to derive one from the other?
Anarchasis
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Do "typical" QFT's lack a lagrangian description?

Sometimes as a result of learning new things you realize that you are incredibly confused about something you thought you understood very well, and that perhaps your intuition needs to be revised. This happened to me when thinking about…
Surgical Commander
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Can Lagrangian be thought of as a metric?

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final configuration as being the shortest one? Then the equations…
N. Virgo
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4 answers

How do I show that there exists variational/action principle for a given classical system?

We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's principle) and even General Relativity (we get…
Debangshu
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Noether's current expression in Peskin and Schroeder

In the second chapter of Peskin and Schroeder, An Introduction to Quantum Field Theory, it is said that the action is invariant if the Lagrangian density changes by a four-divergence. But if we calculate any change in Lagrangian density we observe…
excitedaboutphysics
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On a trick to derive the Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the variation local: the standard argument, which doesn't…
jj_p
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