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).

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1answer
36 views

Confusion about virtual displacements

I am self-studying Goldstein's book "Classical Mechanics", and I need some help understanding the part where Goldstein discusses using Hamilton's principle to solve systems with holonomic constraints (...
6
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2answers
90 views

Actions that are not integrals

So far every action I've seen in physics has been an integral of a Lagrangian, be it a point particle: $$S = \int dt\ L$$ or fields (relativistic or not): $$S = \int d^4x\ \mathcal{L}$$ and so ...
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3answers
204 views

Principle of Least Action Question

Let's say we have a particle with no forces on it. The path that this classical particle takes is the one that minimizes the integral $$\frac{1}{2}m\int_{t_i}^{t_f}v^2dt.$$ So if we graph this for ...
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1answer
110 views

Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
1
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1answer
74 views

Euler-Lagrange Equation in Quantum Field Theory

The quantum fields are operator valued distributions. In some sloppy books like Peskin and Schroeder the Euler-Lagrange equation are used to get the equations of motion. What does it mean to take a ...
7
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1answer
152 views

Canonical second quantization vs canonical quantization with multisymplectic form in AQFT

First of all, I'm a mathematician that knows less than the basics of QFT, so forgive me if this question is trivial. Please, keep in my mind that my background in physics is very poor. 1) The usual ...
4
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1answer
300 views

Equations of motion of displacement field

We have an action: $$S[\boldsymbol{u}] = \frac{1}{2} \int dt \int d^3x \left\{ \mu (\frac{\partial u_{i}}{\partial t})^{2} - \nu (u_{ii})^{2} - \rho(u_{ij})^{2}\right\} $$ Where $u_{ij} = (\partial_{...
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0answers
31 views

Functional variation of potential in integral form

I am trying to vary the following action, $$ S=\int_{t_0}^{t_1} \text{d}t\,(v^\mu v^\nu g_{\mu\nu} + V(t)) =\int_{t_0}^{t_1}\text{d}t\,(v^\mu v^\nu g_{\mu\nu} + \int_{t_0}^t\text{d}s T_\mu v^\mu) $$ ...
3
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0answers
44 views

Lagrangian of classical electromagnetism without $A_{\mu}$ field [duplicate]

Is there a Lagrangian reproducing Maxwell's equations without the use of the scalar and vector potential? Obviously $\mathcal{L} = -\frac14F_{\mu \nu}F^{\mu \nu}$ doesn't work since in terms of $E$ ...
7
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4answers
838 views

Least-action classical electrodynamics without potentials

Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which ...
1
vote
1answer
41 views

Question about Fermat's principle

Why when deriving the law of reflection from Fermat's principle of least time do I set $dL/dx = 0$? I am a 12 grade student with a little notions of maxima and mimima in one variable calculus.
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2answers
1k views

Functional derivative in Lagrangian field theory

The following functional derivative holds: \begin{align} \frac{\delta q(t)}{\delta q(t')} ~=~ \delta(t-t') \end{align} and \begin{align} \frac{\delta \dot{q}(t)}{\delta q(t')} ~=~ \delta'(t-t') \end{...
0
votes
1answer
48 views

Equation of string as hamiltonian field equations

I am using Hamiltonian field theory for the first time and I struggle with some final steps. The task is to derive the equation of vibrating string using Hamilton's field equations. Here is what I ...
4
votes
1answer
99 views

Lagrangian density for Lorentz force of continuous charge distribution in external field?

It's frequently an exercise to derive the Lorentz force law for a particle with charge $q$ in an external electromagnetic field given by the following Lagrangian: $$L = -mc^2\sqrt{1-\frac{\dot{r}^2}{...
0
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2answers
56 views

Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
8
votes
1answer
121 views

Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
7
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3answers
218 views

Can Schrödinger Equation be derived from Huygens' Principle?

Notes of Enrico Fermi start from an analogy between mechanics and optics and with 4 pages he derives the Schrödinger equation. In all my courses, I have seen as an axiom - this is how wave-particles ...
5
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1answer
595 views

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 ...
0
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0answers
43 views

What to do when the fields are arranged in a matrix?

I am dealing with a Lagrangian in which the fields are arranged in an $N\times N$ matrix and i have to find the minima of the potential. Usually i would write the Lagrangian in components and then ...
3
votes
1answer
128 views

Why one should follow Snell's law for shortest time?

whenever two media and two velocities are involved, one must follow Snell's law if one wants to take the shortest time. Why snells law must be followed to travel diffrent media in shortest time? ...
1
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1answer
106 views

Polyakov From Nambu-Goto Directly, for Strings?

The following derivation, for a classical relativistic point particle, of the 'Polyakov' form of the action from the 'Nambu-Goto' form of the action, without any tricks - no equations of motion or ...
5
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1answer
210 views

How is Fermat's least time principle proven?

How is Fermat's least time principle proven? Or it is what usually is observed and is basis for the theories?
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2answers
2k views

Hydrostatic friction: why do water droplets stay at rest on an inclined glass surface?

This is a non-expert question on a (seemingly simple) text-book topic. The question is about "hydrostatic friction", defined as follows. Consider a drop of water resting on a flat surface. If the ...
4
votes
1answer
670 views

Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
23
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4answers
1k views

The Lagrangian 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 ...
16
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4answers
2k views

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 ?
0
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1answer
76 views

How are Lagrangians in QFT constructed?

Various particle equations (like the K-G equation, the Dirac equation, the Proca equation etc.) in QFT are derived by applying the Euler-Lagrange equations to the Lagrangian density. But how are these ...
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2answers
59 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 $t_2$ ...
11
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0answers
153 views

What's the lowest nuclear charge $Z < 1$ that will support a bound two-electron ion $(Z,2e^-)$?

In my programming project I calculate the minimal energy of an atom with 2 electrons in the $L=0, S=0$ state, using a Hylleraas wave function. The values I find for $Z=2$ (He) and $Z=1$ (H$^-$) are ...
4
votes
2answers
86 views

Total derivatives in GR

Without gravity we can easily switch between terms in a Lagrangian, such as $\partial\phi\partial\bar{\phi}$ and $\phi\Box\bar{\phi}$, since total derivative vanishes. But in GR we have additional $e\...
4
votes
1answer
190 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
8
votes
1answer
142 views

When is numerical value of Lagrangian evaluated on-shell a full differential?

I noticed recently that for many field equations, Lagrangian evaluated on-shell (i.e. using equations of motions) is a full derivative- a divergence or something, or in other words a boundary term. ...
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4answers
1k views

Lagrangian for relativistic massless point particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~\sqrt{ g_{\mu\nu} \dot{x}^{\mu}\dot{x}^{\nu}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of ...
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2answers
80 views

Derivation of Euler-Lagrange equation from principle of least action

When deriving the Euler-Lagrange equation for a field $\phi$ the term $$ \int\textrm{d}x^{\mu}~\partial_{\mu}\left( \dfrac{\partial \mathcal{L} }{\partial(\partial_{\mu}\phi)}\right)\delta\phi $$ is ...
7
votes
2answers
956 views

Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
0
votes
1answer
187 views

Acoustical wave equation from Hamilton's principle

It is common to show the features and power of the Hamilton's principle by deriving the equation of vibrating string, membrane etc. using this principle. But I have never seen that used for deriving ...
17
votes
2answers
576 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
0
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0answers
67 views

Physical motivation for Lagrangian formalism

This is more of a request for clarification of understanding and intuition rather than a question, but I hope people can help me with it. I have learned calculus of variations and have subsequently ...
2
votes
1answer
541 views

Variation of the metric with respect to the metric

For a variation of the metric $g^{\mu\nu}$ with respect to $g^{\alpha\beta}$ you might expect the result (at least I did): \begin{equation} \frac{\delta g^{\mu\nu}}{\delta g^{\alpha\beta}}= \delta^\...
4
votes
2answers
98 views

Higher than Lagrangian/action?

When you begin learning physics, you start with equations of motion applied to various physics systems. In classical mechanics course you learn, that exists Lagrangian/action of a system, which gives ...
0
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0answers
123 views

Simple real life applications of Euler-Lagrange equations of motion

If you read some introductory mechanics text like David Morin's Introduction to Classical Mechanics about Euler Lagrange Equations you get a large amount of simple examples like the "moving plane" (...
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0answers
28 views

Must there exist a Lagrangian for any 2nd order ordinary derivative equation?

We know if there exist a Lagrangian of some ODE, then it must exist many equivalent Lagrangian. My question: Then must there exist a Lagrangian for any 2nd order ODE? If not, do we have some ...
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0answers
34 views

Is energy always a diminishing quantity in imaginary time?

If we write Schrodinger equation in imaginary time $\tau = it$, then one can easily show that the energy $E(\tau) = \langle \psi(\tau)| \hat{H} |\psi(\tau)\rangle$ is a diminishing quantity, i.e. $$\...
3
votes
1answer
112 views

Is there an action for every physical law?

Given an action, I can get the differential equation governing the evolution of the system by applying the principle of least action. Does it work the other way around? Given any differential ...
0
votes
1answer
55 views

Understanding Derivation of Euler-Lagrange

I am trying to understand the derivation of the Euler-Lagrange equation. I drew a graph below. So, according to the graph, $$ \int_{t_1}^{t_2} L(x+\delta{x},\dot{x}+\delta\dot{x}\,t) dt - \int_{...
7
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2answers
501 views

Time-dependent Schrodinger equation from variational principle

In the paper, "Density-functional theory for time-dependent systems" Physical Review Letters 52 (12): 997 the authors mentioned that the action $$ A= \int_{t_0}^{t_1} \mathrm dt \langle \Phi(t) | i \...
3
votes
1answer
189 views

Hamilton's equations from the action with boundary conditions involving position and momentum

Generally, when you are given the action $$ S=\int_{t_1}^{t_2}\mathrm dt (p\dot q - \mathcal H )$$ the boundary conditions are $q(t_1)=q_1$ and $q(t_2)=q_2$. This is useful because to calculate $\...
3
votes
1answer
84 views

Why can you make $V$ stationary with respect to a parameter of the field in Derrick's theorem?

I'm going over Coleman's derivation of Derrick's theorem for real scalar fields in the chapter Classical lumps and their quantum descendants from Aspects of Symmetry (page 194). Theorem: Let $\...
1
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2answers
433 views

Derrick’s theorem

Consider a theory in $D$ spatial dimensions involving one or more scalar fields $\phi_a$, with a Lagrangian density of the form $$L= \frac{1}{2} G_{ab}(\phi) \partial_\mu \phi_a \partial^\mu \phi_b- ...
3
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0answers
85 views

On the surface, is the law of maximum entropy production the same as principle of least action?

I just have read about the law of maximum entropy production. Someone has idolized it enough to make an whole website just for it: http://www.lawofmaximumentropyproduction.com/ A system will ...