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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Action functional in the formalism of symplectic manifolds with Hamiltonian
Call Hamiltonian a symplectic manifold $(M, \omega)$ equipped with a distinguished Hamiltonian $h \in \mathcal C^\infty(\mathbb R \times M)$.
Wikipedia 'Tautological form' page has a section about ...
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Can the full set of II. Bianchi identities be derived from the symmetries of the action?
In pseudo-Riemannian geometry we can derive the II. Bianchi identities by considering, e.g. the expression of the Riemann tensor in Riemann normal coordinates. They read
$$R_{\mu\nu\kappa\lambda;\...
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Is action principle trivial? [duplicate]
Given a function $f(t)$, is it possible to construct Lagrangian $\mathcal{L}$ such that
$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial\mathcal{L}}{\partial \dot{f}}=\frac{\partial\mathcal{L}}{\partial ...
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1answer
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Independence of position and momentum in action
Why are position and momentum independent with respect to the Hamiltonian Action $S_H$ given by
$$
S_H = \int_{t_1}^{t_2} (p \dot q - H) dt \ \ \ ? \tag{1}
$$
While deriving Hamilton's equations from ...
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1answer
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Why Lagrangian has this form in general relativity?
One can derive the geodesic equation by Euler-Lagrangian equation,
\begin{equation}
\dfrac{\partial \mathcal{L}}{\partial x^\gamma} - \dfrac{d}{ds}\bigg(\dfrac{\partial \mathcal{L}}{\partial (dx^\...
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Other infinitesimal variation of the action
I was reading this post about the virial theorem where the virial theorem comes from varying the action by the infinitesimal rescaling $x\rightarrow(1+\epsilon)x$ and asking that $\delta S=0$ under ...
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Geodesics from a Lagrangian in a restricted space
Given a certain action (for instance),
\begin{equation}
S = \int_\alpha^\beta d\lambda ~ L(\dot x, x, \lambda)
\end{equation}
where $\lambda$ is some affine parameter.
In order to minimise it, we ...
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Significance of Lagrangian in Principle of Least Action?
I've been studying the Legendre transform and it's been a fun realization to see that the relationship between the Lagrangian and the Hamiltonian is simply a Legendre transform, i.e.,
$$\{H, p\}\...
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What will happen if we take up Cartesian Coordinates in Lagrangian Formulation instead of Generalized Coordinates?
Why do we actually need generalized coordinates? Is it a mathematical manipulation only or does it serve physical purpose? And will principle of stationary action stay valid if we use cartesian ...
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Time reversal on potential $V(\frac{d^n {\vec x}( t)}{d t^n})$ with any odd or even power time-derivative on the position function
In this post, I tried to challenge what @Richard Myers said in his answer in https://physics.stackexchange.com/a/633205/42982. I followed what he said, except I kept a common widely used notation $\...
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The first and second form of Euler-(Lagrange) equation with explicit time dependence
I have learned the first and second form of Euler-(Lagrange) equation with no explicit time dependence (the time dependence only implicit on the function to be solved, say $y\left(t\right)$), from ...
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Functional derivative or Euler-Lagrange? [closed]
What is the difference between the functional derivative and the Euler-Lagrange equation?
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Lagrange equation 1788, and Hamilton principle 1834
Lagrange's equation and Lagrangian and derived in 1788. It is different from Newtonian mechanics view because Newton emphasizes the external force acts on the body. But the Lagrange's. view is that ...
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1answer
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Solve the hanging rope shape using the variational principle? [duplicate]
I know how to write down the local ordinary differential equation (ODE) via Newton's force law, balancing between the left and right rope tension and the gravity exerted on a infinitesimal piece of ...
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Does universal speed limit of information contradict the ability of a particle to pick a trajectory using Principle of Least Action?
I'm doing some self reading on Lagrangian Mechanics and Special Relavivity. The following are two statements that seem to be taken as absolute fundamentals and yet I'm unable to reconcile one with the ...
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Can we go from equations of motion back to Lagrangian? [duplicate]
We always go in one direction, from Lagrangian to equations of motion (classical mechanics). But is it possible to go the opposite way, from equation of motion to Lagrangian? Suppose we have an ...
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Does every Newtonian system have an Euler-Lagrangian counterpart (possibly time-dep, no generalized forces)? Or is there an explicit counterexample?
Given a Newtonian system (with forces that may depend position, velocity, and time) with solutions $\vec{q}(t)$, can we always derive a Lagrangian system such that the following are satisfied:
It ...
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What's special about the $T-V = H- 2V =2 T- H$, which the least action principle extremizes? [duplicate]
My question is simple.
In the classical mechanics, we know the least action principle does the variation on the action
$$
S = \int (T-V) dt ,
$$
where $T$ is the kinetic term and $V$ is the potential ...
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1answer
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How to prove that there is only trajectory given fixed boundary conditions, if you know the Lagrangian of the system?
In my particular problem, the Lagrangian of the system is:
$$
L = \frac{m(\dot r^2 + r^2\dot \varphi^2)}{2}
+ \frac{m\omega^2 (r\sin \varphi)^2}{2}
$$
From there, we can derive the equations ...
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2answers
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Carroll: Energy-momentum tensor for a scalar field theory
In Carroll's Introduction to General Relativity: Spacetime and Geometry, there is a section titled Classical Field Theory in chapter 1. There, he mentions that:
"The action leads via a direct ...
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Is it possible to show simply using the Lagrangian that a body in free fall (& $v_i=0$) follows the most “ efficient” path ( i.e. a vertical line)?
In this video lecture by M. van Biezen ( Loyola Marymount Uni) https://www.youtube.com/watch?v=uFnTRJ2be7I&list=PLX2gX-ftPVXWK0GOFDi7FcmIMMhY_7fU9&index=2
it is shown how to apply the ...
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3answers
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Least action principle remark on negative mass in Landau-Lifshitz classical mechanics
In the famous Landau-Lifshitz's Classical Mechanics there is a remark I cannot fully understand at the very beginning of the book (page 7 of the second edition):
It is easy to see that mass of a ...
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1answer
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Why does it seem like there is always a Lagrangian? [duplicate]
All the fundamental laws of physics can be written in terms of an action principle. This includes electromagnetism, general relativity, the standard model of particle physics, and attempts to go ...
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2answers
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Relation between self-adjointness and variational principle and Rayleigh's principle
In mathematical physics, why is it that when an eigen-equation is described by a self-adjoint operator we say that it can be written (formulated) as a variational action (or principle)?
Does the ...
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2answers
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Do dynamic systems that are based on a variational principle imply a conservation law?
In many dynamic systems in classical physics, as well as quantum mechanics, the equation of motion can be derived from a variational principle (VP), i.e. minimizing an action integral of some sort.
I ...
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Why does nature favour systems that follow from a variational principle? [closed]
When Newton discovered ‘Newton’s law’ he was probably not aware that it could be viewed as a consequence of minimizing an ‘action integral’ (integral of some Lagrangian density).
Since the same is ...
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Mathematical expression for refraction within a spherical lens
Qs: How do I show that for a point object, for a ray at a large angle from the optical axis, spherical aberration (SA) produces a distorted focal length $f_{SA}$ that is shorter than the normal focal ...
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Is this quantum mechanical proof of the virial theorem general?
I have seen the following proof for the virial theorem in QM using the variational method. It goes like this:
Suppose an exact eigenstate of the system is $\psi(\vec{r})$ and consider a variational ...
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Why does the curve of a hanging chain not minimize the area below it?
If we have a chain of fixed length hanging from two points we know that it will form a curve that minimizes the chain's potential energy.
If we imagine the chain as having many small segments, then ...
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How does principle of least time suggest a relation between three indices of refraction?
In the 26th Feynman Lecture, Fermat's principle of least time is discussed and this point about refractive index is brought up:
It is easy to show that there are a number of new things predicted by ...
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Global conserved quantities for point particle coupled to a Schrodinger field
We have a box (represented by a potential $V$) with a classical particle in it. If the box has a finite inertia and it's floating in space, then it shakes as the particle bumps on the walls. The total ...
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GR Lagrangian with higher-order curvature terms
I'm trying to find papers / books / lectures with the derivation of the equations of motion from Lagrangians with higher order in curvature terms, for example with the Kretschmann scalar $R_{\mu \nu \...
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Connection between different kinds of “Lagrangian”
Being a physic student I first heard the term: "Lagrangian" during a course about Lagrangian mechanics; at that time this term was defined to me in the following way:
For a classic, non ...
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Four-Dimensional Action Principle for a free particle
Following Landau & Lifshitz "The Classical Theory of Fields" chapter 2, p. 27, according to the principle of least action, we have (for a free particle):
$$ \delta S = -mc \delta \int^...
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1answer
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Must the action be a coordinate scalar?
I know that an action must be locally-Lorentz invariant based on physical reasons, but is there any requirement for it to be a coordinate pseudo-scalar (up to surface terms)? In particular, would an ...
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1answer
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How does the approximate energy relate with the trial function in variational method?
While studying the variational principle in Quantum Mechanics, I came across the following problem: I want to relate the difference between the exact ground-state wave function $\psi_0$ and an ...
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2answers
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Jerk mechanics - Lagrangian
I have a Lagrangian with the form
$$L = L[q(t,\alpha), \dot{q}(t,\alpha), \ddot{q}(t,\alpha), t],$$
to which I am applying the calculus of variations. The problem is that when I apply the calculus, I ...
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1answer
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Endpoints in Fermat's Principle
Are the endpoints of the light ray path in Fermat's principle must be fixed?
To clarify my question: Using Wikipedia definition for Fermat's Principle:
Fermat's principle states that the path taken ...
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Reversibility of light rays, Faraday effect and Fermat's principle
I am currently having some struggle to understand the connection between the following three concepts in optics:
Law of Reversibility (Geometrical Optics): The direction of light rays does not matter ...
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Derivation of Fermat's Principle from the Least Action Principle [duplicate]
Can I prove that Fermat's Principle (i.e. that light beam travels between two points in the path that require minimum time) using the Least Action Principle for electromagnetic wave? (from analytical ...
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A mathematical step done by Schrödinger in his solution to the hydrogen atom I need clarification on [duplicate]
in his paper "Quantisierung als Eigenwertproblem" he begins by introducing the Hamilton-Jacobi equation:
$$H\left (q,\frac{\partial{S}}{\partial{q}} \right)=E\tag{1}$$
such that $S$ is ...
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Symmetry principles and analogous Noether's theorem for stochastic systems
In the case of Newtonian mechanics, taking a variational or Lagrangian or Maximum principle (MP) view, one can obtain the conservation laws (energy, linear momentum and angular momentum) by combining ...
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When to use (and when not to use) electromagnetic field conjugates in variational formulations
I found something a little bit confusing about writing variational formulas or Lagrangians for electromagnetic fields. I was looking at the book by Schwinger and Milton (chapter 4), and saw that ...
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Worldline action of point particle in gravitational field
In my GR lectures on the derivation of geodesic equations via extremal length, my lecturer wrote that the worldline action $S$ of a point particle with mass $m$ is given by $$S=-m\int\sqrt{-g_{\alpha\...
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Least Action in General Relativity
For an affinely parameterised geodesic we can form the Lagrangian:
$$
\mathcal L = g_{ab}\dot x^a\dot x^b = \text{constant}
$$
The Lagrangian is constant by the fact that the geodesic parallel ...
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1answer
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Can you write and solve the EM field Lagrangian density without reference to the EM potential? [duplicate]
Is it possible to write the Lagrangian density for the EM field and charges
$$L=-\frac{1}{4\mu_0}F^{\mu \nu}F_{\mu \nu}+j^{\mu}A_{\mu}$$
only in terms of the Electromagnetic Tensor and current vector? ...
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Lagrangian stationarity to find amplitude of perturbed system eigenfunction
Say we have a system that satisfies the Helmholtz wave equation $\nabla^{2}h+k_{0}^{2}h=0$ and some boundary conditions over the boundary of the system volume $V$ in 3D space. Let the field solution ...
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Is there a minimization principle for Hamiltonian? [duplicate]
Consider a point particle in $n$ dimensions.
For a Lagrangian $\mathcal L(\mathbf{q, \dot q}, t)$, we have that $\mathbf q(t)$ is a feasible trajectory for times $t_0<t<t_f$ iff it extremizes ...
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1answer
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How is the relationship between the old and the new canonical variables justified?
In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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Hamilton's principle for fields
According to Goldstein, Hamilton's principle can be summerized as follows:
The motion of the system from time $t_{1}$ to time $t_{2}$ is such that the line integral (called the action or the action ...
