New answers tagged lagrangian-formalism
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Are there any experiments that examine Hamilton's Principle directly?
About principles in physics:
I want to refer to a discussion by stackexchange contributor Kevin Zhou.
There is a Januari 2020 question titled: 'Why can't the Schrödinger equation be derived?'
In his ...
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Why doesn't Car-Parrinello molecular dynamics require an SCF calculation?
It is not entirely true that Car-Parrinello (CP) Molecular Dynamics (MD) doesn't require a self-consistent field (SCF) calculation at all. The method must start close to the SCF solution (actually a ...
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When is the Lagrangian a Lorentz scalar?
An obvious, kind of dumb, answer is that the Lagrangian corresponding to a given Hamiltonian will be a Lorentz scalar if the Hamiltonian has the form,
$$ \mathcal{H} = \pi^a \frac{\partial}{\partial t}...
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Are there any experiments that examine Hamilton's Principle directly?
Often in theoretical physics, there can be a large gap between the logical starting point of a theory, and the actual experimental tests.
The principle of least action (aka Hamilton's principle) is ...
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When is the Lagrangian a Lorentz scalar?
As far as I know, there are no good ways of stating what conditions on the Hamiltonian will cause the Lagrangian to be a Lorentz scalar other than to just say the Hamiltonian must be derived from a ...
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Is First-Class Constraint Generator of matter Gauge Symmetry in EM example?
The solution to OP's problem is to include a pertinent matter sector in the E&M Lagrangian (3) (in OP's case: a complex scalar $\phi$). This produces the source term in the first-class secondary ...
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Is First-Class Constraint Generator of matter Gauge Symmetry in EM example?
From @Andrew's comment, I know how to solve it.
In scalar QED, EM field would couple to current $J^{\mu}(x) = \phi^{\star}(x)\partial^{\mu}\phi(x) - \phi(x)\partial^{\mu}\phi^{\star}(x)$, and the ...
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Relation between energy and time
I understand the question as the OP asking of an intuitive way to think about the relation between energy and time. Therefore the point I would like to make is that energy isn't really a thing.
So for ...
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Relation between energy and time
The way I like to look at it (and which may or may not give you the same amount of intuition as it does to me) is as such:
Momentum is what gives rise to changes in position. Clasically, a body ...
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Relation between energy and time
Why is energy always related to time in physics.
I don't think it is helpful to describe energy as "always related to time" in physics.
That being said, there certainly are a number of ...
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Relation between energy and time
There are at least three occasions where the notions of Energy and Time show up together in classical and modern physics.
Probably the most elementary situation is related to the fact that the ...
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Does quasi-symmetry preserve the solution of the equation of motion?
Boundary condition is a part of the very definition of your field theory, classical or quantum. When e.g Peskin & Shroeder leaves out the surface term, "all fields and derivatives vanish at ...
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Why should an action integral be stationary? On what basis did Hamilton state this principle?
In this answer I will proceed from $F=ma$ to Hamilton's stationary action in forward steps.
Some historical remarks:
In his 1834 paper William Rowan Hamilton described an approach that combined two ...
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Missing counterterms in $\phi^3$ + $\phi^4$ theory in 1PI effective action
First of all, this 1PI thing is quite comparable to perturbation series. Since you use Peskin/Schroeder, check out the reference to that Coleman Weinberg potential project: Coleman and Weinberg in ...
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Energy-momentum tensor and equation of motion in Einstein-Dilaton theory
Given an action
$$S[\psi]=\int d^4x \mathcal{L}(\psi,\nabla_\mu \psi),$$
how do we find the equations of motion? The principle of stationary action says that the equations of motion can be found by ...
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QFT introduction: From point mechanics to the continuum
What do Peskin & Schroeder mean by $\dot{\phi}(\vec{x})$? How can we define a time derivative of a function, which depends on position only?
Notation $\dot \phi$, when writing down Lagrangian/...
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QFT introduction: From point mechanics to the continuum
I know that the four derivative (and the four vector) inside the Lagrangian of (Eq. 2) is crucial to maintain Lorentz invariance, but that is the only thing that is not predicted by the four ...
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How can the stress-energy tensor $T^{μν}$ be unique when the Lagrangian $L$ is not?
If you get $T$ that way from the Lagrangian, it need not be unique. This is just like with Hamiltonian derived from a Lagrangian in ordinary mechanics; it is not unique, because different $L$ (e.g. ...
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How can the stress-energy tensor $T^{μν}$ be unique when the Lagrangian $L$ is not?
In principle, for standard local Lagrangians of field theory, the ambiguity of the Lagrangian density of the matter field $\phi$ amounts to a total derivative (just beacuse it should not change the ...
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
The notation is a little sloppy from a purely mathematical point of view (although common in physics) so it might be causing a little confusion.
To help clarify, it might help to use different letters ...
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
They are partial derivatives. From the chain rule, we have
$$
\frac{\partial V(aq_1-bq_2)}{\partial q_1}= a V'(aq_1-bq_2),\\
\frac{\partial V(aq_1-bq_2)}{\partial q_2}= -b V'(aq_1-bq_2).
$$
For ...
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
No. The notation means that the V on the RHS is a function only of one variable, and so its derivative is the simplest, one-variable derivative.
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
I think it is gibberish in your own words. $\frac{\partial}{\partial \dot{q}}$ cannot be replaced by $dt \frac{\partial}{\partial {q}}$ even in the most cavalier approach because $\dot{q} = \frac{dq}{...
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Meaning of $d\mathcal{L}=-H$ in analytical mechanics?
Apart from the fact that I am really skeptical about its mathematical validity, your replacement $\frac{\partial}{\partial\dot{q}}\rightarrow\frac{dt}{\partial q}$ makes little sense in the context of ...
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Volume preserving transformation in the Circular Restricted Three-Body problem
OP is right that OP's map (1) $$(x,y,p_x,p_y)\quad\stackrel{f}{\mapsto}\quad (x,y,v_x,v_y)$$ is volume preserving
$$ f^{\ast}(\omega\wedge\omega)~=~ \omega\wedge\omega, $$
although not a ...
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How can the stress-energy tensor $T^{μν}$ be unique when the Lagrangian $L$ is not?
Adding a constant to the Lagrangian actually does affect the physics when gravity is involved. This constant term has an interpretation as the energy of the vacuum, or the cosmological constant. For ...
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Equivalence between Hamiltonian and Lagrangian Mechanics
Lagrangian and Hamiltonian mechanics are not exactly equivalent because they do not cover the same possibilities for the system to be described. Actually, just using the Newtonian laws gives yet ...
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Is there an error in Susskinds' derivation of Euler-Lagrange equations?
Here are my thoughts (I may be wrong). The first equation states the action which is the integral of the Lagrangian with respect to time. Integrating is the equivalent of finding the areas of strips, ...
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
OP asks an interesting question, which we rephrase as follows:
Given an action
$$S[q]~=~\int_{t_1}^{t_2} \!dt~L(q,\frac{dq}{dt},t),$$
are the stationary action principle (SAP) and Euler-Lagrange (EL) ...
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Variation in the context of symmetries
The Einstein-Hilbert action is
$$ S = \frac{1}{2\kappa} \int R \sqrt{-g} d^4x$$
with $\kappa$ the dimensionful constants.
If you set $R=0$ at the outset, you have the action $S = 0$, which is a ...
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Variation in the context of symmetries
It's hard to say for sure, since you don't remember the details, but usually we don't assume anything about the solutions to the equations of motion when deriving them (by computing the variation of ...
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Does Hamilton's principle allow a path to have both a process of time forward evolution and a process of time backward evolution?
As I understand it you are wondering about the following:
What will be the implications when Hamilton's action is evaluated over a time interval where the end point is earlier than the start point?
I ...
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
If you are going to differentiate $L$ with the respect to the metric, $L$ needs to be rewritten without the metric being implicitly used anywhere. Otherwise, you will not vary the entire dependence on ...
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Action-angle variables for three-dimensional harmonic oscillator using cylindrical coordinates
There are some issues in your computation. It seems that you assume that you will be able to use identities from the 1D harmonic oscillator to solve for AA coordinates in this more general case. This ...
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Which higher-order terms require 4th-order integration of quadratically-constrained dynamics?
You need to be careful in applying leapfrog to your EoM's since you have a velocity dependence on the force. You can have an appropriate second-order solver if you derive it using the same principles.
...
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Possible boundary conditions in derivation of Euler-Lagrange equations
As mentioned in the linked Phys.SE post there are 2 possible boundary conditions (BCs):
Essential/Dirichlet BC,
Natural BC,
which can be chosen independently for each connected components of the ...
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Which higher-order terms require 4th-order integration of quadratically-constrained dynamics?
To answer the titular question: there isn't such a rule. Effectively, it's up to you as the researcher to determine the best ODE scheme to use to solve the problem based on your wants.
For instance, ...
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Vanishing virtual work done by non-holonomic constraints
Ref. 1 is effectively considering $m$ independent non-holonomic constraints of the form$^1$
$$ f_{\ell}(q,\dot{q},t)~\equiv~\underbrace{g_{\ell}(q,\dot{q},t)}_{=\frac{1}{n}\dot{q}^j\frac{\partial g_{\...
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Derivation of Noether Current in Condensed Matter Field Theory by Altland and Simons
A&S assumes that the action has a strict$^1$ (rather than a quasi-)symmetry, and so that the bare Noether current (1.43) without improvement terms suffice.
--
$^1$ NB: It is implicitly assumed ...
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