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).
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Is Hilbert-Einstein action just the leading order of some kind of series?
Introducing the action for the gravitational field my GR professor stated that, in principle, one could write it as
$$S = k\int d^4x\sqrt{g}(\sum_n\sum_mR_n^m - 2\Lambda), \space \space \space \space ...
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42 views
Time-independent Schrödinger equation Lagrangian derivation
Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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25 views
How general is the Lagrangian formulation? [duplicate]
Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers.
I've been ...
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2answers
82 views
Euler-Lagrange equations from a complex Lagrangian
I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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0answers
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Euler applies the principle of least action to projectile motion
I was reading my physics book and became interested in the origin of the Principle of least action. After some historical research I arrived at Euler's historical text A method for finding curved ...
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2answers
134 views
Is Action Always “Locally” Least?
In general, I know it's true that the Principle of Least Action is more properly called the Principle of "Stationary" Action. However, there are results which seem to suggest that for sufficiently ...
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1answer
60 views
Proving that the Euler-Lagrange Equation has no solution [closed]
I'm trying to show that the Euler - Lagrange equation for the functional $$I(y)=\int_{a}^{b} y\:dx$$
subject to $y(0)=y(1)=0$ has no solutions.
The Euler - Lagrange equation states that:
$$\frac{d}{...
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37 views
Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]
Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
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43 views
Equations of motion from action variation
I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action:
$$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
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2answers
56 views
How does Hamilton's Principle give us the path taken?
We defined the action as:
$$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$
where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed.
Hamilton's principle states that the path that is ...
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1answer
57 views
How can Lagrangian method work whenever the Lagrangian is not convex?
Let $$L(x,\dot x)=\frac{1}{2}m\dot x^2-\frac{1}{2}k(x-x_0)^2-mgx$$ the Lagrangian of a system. Euler Lagrange theorem says that a necessary condition to be a minimizer is to satisfy Euler-Lagrange ...
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2answers
128 views
History of Principle of Least Action [closed]
I am interested in how Lagrange came up with Principle of Least Action.
Is it derived from some experimental data or by mathematical deduction? And how? Either seems hard.
I hope this might ...
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1answer
19 views
Lagrange's equation for system not having time translation
While we are deriving Lagrange's equation from D'Alembert's principle, when we argues as;
$$\delta \vec r_\alpha = \sum_i \frac{\partial \vec r_\alpha}{\partial
q_i }\delta q_i + \frac{ \partial \...
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0answers
22 views
Deriving Canonical Transformation from Generating Function using Principle of Stationary Action
In Hamill's "A Student's Guide to Lagrangians and Hamiltonians", section 5.2, the equations for a canonical transformation $(q,p) \to (Q,P)$, induced by the generating function $F(q,Q,t)$ are derived ...
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1answer
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Attaining extrema when a stationarity condition has no solution
I was wondering if someone could shed some light on the following for me:
If a stationarity (maximizing or minimizing) condition has no solution inside a particular domain, then how do we reason that ...
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2answers
55 views
Paths of least action and loops in time
In the book
Quantum Field Theory for the Gifted Amateur
link: https://books.google.ca/books?hl=en&lr=&id=nIk6AwAAQBAJ&oi=fnd&pg=PP1&ots=JZjwG_qDt5&sig=...
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0answers
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Need help with Gauss Bonnet gravity
I am trying to find the equations of motion for a coupling Gauss-Bonnet gravity.
The action is :
\begin{equation}
S=\int{d^4x \sqrt{-g}\left[\frac{R}{2k}+f(\phi)(\alpha R^2+\beta R_{\mu\nu} R^{\...
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6answers
222 views
Why relativistic Lagrangian doesn't simply equal kinetic minus potential energy $L=T-V$?
As the question above, I wonder why the relativistic Lagrangian is written as:
$$L=-mc² \sqrt{1-\frac{v²}{c²}} - V ~=~-\frac{mc^2}{\gamma} -V~?$$
I know that the kinetic energy of a relativistic ...
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0answers
38 views
Transformation of ADM parameters under diffeomorphisms
I am trying to prove the invariance of the ADM formalism under (infinitesimal) diffeomorphisms. I have checked Wald and other textbooks on the subject but have been unable to find expressions for how ...
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0answers
36 views
Lagrangian for first order equation of motion? [duplicate]
Let us have the following equation of motion (it might not necessarily correspond to a physical system):
$$\dot{x} + a \cdot x + b \cdot x^2 + c=0.$$
I would like to deduce the corresponding ...
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0answers
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How can we predict how a system evolves using the stationary action principle even though we need to specify the final state? [duplicate]
The stationary action principle states that a system evolves between a fixed initial and fixed final configuration in such a way that the action is stationary.
But isn't the final configuration what ...
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0answers
49 views
Rigid Body Equations in terms of Body Coordinates by Hamilton's Principle
I sought-for the equations of motion of an unrestrained rigid body. The equations of motion are readily available in the literature, but my concern is to derive them by Hamilton's principle.
...
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2answers
75 views
Euler-Lagrange Equation, Shortest Path On Sphere
The equations for spherical polar coordinates are
$$x = r \sin(\theta) \cos(\phi) \\ y = r \sin(\theta) \sin(\phi) \\ z = r \cos(\theta)$$
Now, consider a path expressed as $\phi = \phi(\theta)$...
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2answers
75 views
Approximating ground-state energy without using variational principle
Given the Hamiltonian for one dimension harmonic oscillator: $$H=-\frac{\hbar^2}{2\mu}\frac{d^2}{dx^2}+\frac{\mu\omega}{2}x^2 ,$$ I need to calculate the approximate ground state energy using the ...
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2answers
51 views
Fermat's principle: when does light actually travel along the local maximum of accumulated phase? [duplicate]
In class we learned that Fermat's principle dictates that light travels either along a local minimum or a local maximum of the accumulated optical phase, but the professor only gave examples of local ...
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0answers
35 views
Is Entropy a monotonically increasing function of Gibbs Free Energy/ Helmholtz free energy/ Enthalpy?
Entropy can be axiomatically taken as a monotonically increasing function of internal energy $(E) ,$ from where "energy minimum principle" can be deduced, and this can be stated using variational ...
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1answer
71 views
D'Alembert's principle and equation of motion
Is obtaining proper equation of motion from D'Alembert's principle a mere coincidence or there is some logic behind this?
This is asked because while we are finding the equations in a regime where ...
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4answers
207 views
What is the physical content of the principle of least action?
Say the world is governed by the Principle of Least Action (or Hamiltonian mechanics) and let's not worry about quantum mechanics too much.
Independently of any Lagrangian or Hamiltonian, does that ...
2
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1answer
49 views
Scaling Problem with Variational Method
$\def\braket#1{\langle#1\rangle}$
I am attempting to solve a particular Hamiltonian by variational method. The wavefunction that I have selected is as follows:
$$
\Psi = Ne^{\frac{-kr}{2}}\sum_{i=0}...
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0answers
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Physical Meaning of the Gutzwiller Constraints
I have a doubt on the constraints for the expecation values obtained by Bünemann et all.
First i want to introduce my notation
To analytically solve a tight-binding model,
\begin{equation}
\hat{H}= ...
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1answer
55 views
Variation of action for massive point particle (pp)
So I'm pretty sure I'm missing something obvious, but for the life of me I cannot replicate the step between 1.2.2 and 1.2.3 in Polchinski Vol 1. Basically, I'm trying to find the variation of:
$$S_{...
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1answer
44 views
How can the action can describe a movement? What is the argument behind? [duplicate]
We define the action of a system as $$S(q)=\int_{t_1}^{t_2}L(t,q(t),q'(t))dt,$$
where $q(t)$ is the evolution of the system and $L$ is the Lagrangien. How can a stationary point of $S$ can describe ...
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0answers
20 views
Geodesic curve definition [duplicate]
Do we have a choice in defining the covariant derivative by the use of a set of coefficient functions(Christoffel gammas)? If so, could we then say that these coefficient functions need not to ...
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1answer
47 views
How does Fermat's principle of least time come from this statement? [duplicate]
In Wikipedia Fermat's Principle is stated as:
A ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the ...
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2answers
65 views
Geodesic equations from action with auxiliary field
A textbook says that the geodesic equations (for both massive and massless) can be derived from the following action:
$$
S = -\frac{1}{2} \int d\tau \:\eta \: (\eta^{-2} \dot{x}^\mu \dot{x}^\nu g_{\...
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If there is no destination does light actually get emitted? [closed]
If the action of least principle says light needs a destination what happens when we shine a laser into the dark region of the microwave background radiation?
Alternatively, if we put a light source ...
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1answer
53 views
Least action in differential steps
It says here that the entire integral is minimum if and only if the actions for each step is minimum but here is a contradiction. Suppose there are two fixed points in space, time: A and C. it takes ...
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3answers
56 views
Why does light go in the path where time is minimum and not maximum?
On deriving Snell's Law from Fermat's Principle there is a part where
$\frac{ds}{dt}=0$ where $s$ is the distance gone by light.
But the principle states that light takes the path where it takes ...
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1answer
65 views
Feynman Lecture Principle of Least Action: Glossed over Taylor expansion?
His initial one dimensional derivation of Newton's Second Law using the Principle of Least Action, I believe is fairly concise and easy to read. However, I did get hung up on his use of the Taylor ...
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1answer
95 views
Lagrangian of free particle - classical case
I have a question, more related to a mathematical aspect of physics, which seems I am not understanding very well.
So, by applying Galilean transformation between two reference frames, which move at ...
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2answers
87 views
Why the Lagrangian of a free particle cannot depend on the position or time, explicitly?
On p. 5 in $\S$3 pf the book of Mechanics by Landau & Lifshitz, it is claimed that
[...] for a free particle, the homogeneity of space and time implies
that Lagrangian cannot depend on ...
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1answer
15 views
Overconstrained equilibrium binary mixture phase seggregation
Assume a binary mixture, with the component $A$ in concentration $c$ and the component $B$ in concentration $1-c$. The total energy $E(c)$ is thus given by a function of one variable $c$ (if we do not ...
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0answers
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Maxwell Lagrangian where $F$ and its derivatives are the variables (i.e., without replacing $F={\rm d}A$) [duplicate]
The way the electromagnetic Lagrangian is usually constructed is by noticing that the EM fields are always constrained to satisfy ${\rm d}F=0$ (half of Maxwell's equations). We can immediately solve ...
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3answers
111 views
Is it possible for the Action $S$ to *not* have a stationary point?
So the path of an object in configuration space is given by Hamilton's principle, which states that the path which the particle travels on is the one on which the action is stationary:
$$\delta S = \...
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2answers
122 views
Curve for fastest time down a ramp [duplicate]
I came across a physics experiment video showing three balls released from a point A, going down three different kinds of ramps leading to a point B (https://www.youtube.com/watch?v=61S0KW7e-rc)
...
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2answers
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Example in which light takes the path of maximum optical length [duplicate]
According to the modern version of Fermat's principle,"A light ray in going from point A to point B must traverse an optical path length that is stationary with respect to variations of that path.".Is ...
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2answers
144 views
How to deal with explicit time dependence of the Lagrangian?
Clearly, if the Lagrangian in explicitly time dependent, the Euler-Lagrange equations being satisfied does not extremise the action. I am unclear as to how to deal with systems with an explicitly time-...
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1answer
103 views
Deriving the geodesic equation using a Lagrange multiplier to fix affine parametrisation
The geodesic equation can be derived using the action
$$S_0 ~=~ \int d\tau \sqrt{-\dot{x}_\mu\cdot \dot{x}^\mu}\tag{1}$$
(I am using the (-+++) convention and $\dot{x} = \frac{dx}{d\tau}$). To ...
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1answer
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Polyakov Lagrangian and Lagrange multipliers
I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
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When can we say $x$ and $p$ are “independent variable”, in order to find the Vlasov equation?
I have a question about "independent variable" in physics, and especially variable in Lagrangian or Density Function. I read several questions about it in this forum and although I have the feeling I ...
