Questions tagged [equations-of-motion]
DO NOT USE THIS TAG just because the question contains an equation of motion!
77 questions
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Confusion about Noether's Theorem
In classical mechanics, a transformation $q \rightarrow q + \delta q$ is a symmetry if the resultant change in the Lagrangian is a total derivative,
$$ \delta L = \frac{dF}{dt}.$$
If we derive the ...
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Splitting Scalar into Holomorphic and Anti-Holomorphic Parts
I am reading Tong’s string theory lecture notes. On page 78, he splits the 2d free scalar into left- and right-moving parts, seemingly using the classical equation of motion as justification.
Why is ...
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Gauge invariance using equations of motion [duplicate]
I am working with a lagrangian on a homework problem. I expect it to have some gauge invariance. I can show that the Lagranian is invariant under those (gauge) tansformations but I have to use ...
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Consequences for symmetries of the equations of motion in QFT
In general, if a Quantum Field Theory is described by a Lagrangian $\mathcal{L}$, the symmetries of $\mathcal{L}$ lead to classically conserved currents along the equations of motion and Ward ...
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To find the displacement of a rolling body
When calculating the displacement of a rolling body do we just calculate the displacement due to Vcom in a particular time t or additionally need to consider also the displacement that may be produced ...
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Does the path integral approach to QFT have equations of motion? [duplicate]
In the canonical quantization approach for QFT, we deal with operators & their (anti)commutation relations. However, at the same time, we say that the field operators are the solutions of equation ...
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Why does the ball in Galileo's double inclined plane experiment reach the same height?
Why does the ball in Galileo's double inclined plane experiment reach the same height? I know how to show it by energy conservation law but am unable to prove it by the equations of motion. Can anyone ...
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Noether’s second theorem: about the action principle
Noether's second theorem is supposed to show that the invariance of the Lagrangian by the Lie group (infinite in dimension) of certain theories necessarily implies that the field equations proper to ...
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Why for motion planning of quadrators the goal is to minimize the jerk/snap?
In motion planning for quadrators the optimization goal is sometimes to minimize the (norm squared of the) jerk and more often the (norm squared of the) snap. Can someone provide an intuitive and ...
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Can the $\eta_{\mu\nu}\mathcal{L}$ term in canonical energy–momentum tensor be omitted?
From Noether theory we can define the canonical energy–momentum tensor as
\begin{equation}
T_{\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial(\partial^\mu\phi)}\partial_\nu\phi-\eta_{\mu\nu}\mathcal{...
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Why the classical Euler-Lagrange equation is assumed when deriving the Noether's conserved current?
As known, in QFT, the conserved currents, such as the energy-momentum tensor, can be derived from the Noether's theorem and expressed as the product of the field operators. These conserved currents ...
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Conceptual question about Euler-Lagrange equations in Quantum Field Theory
So I've started going down the QFT rabbit hole aided by Schwartz's book "Quantum Field Theory and the Standard Model". On chapter 7, the first method used to find the position-space Feynman ...
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Using EOM in QED Lagrangian [duplicate]
Let's have the QED Lagrangian.
$$\mathcal{L} = -\frac{1}{4} F_{\mu\nu}F^{\mu\nu} + \bar{\Psi}(i\partial_\mu \gamma^\mu - m)\Psi + g\bar{\Psi}A_\mu \gamma^\mu \Psi.\tag{1}$$
The equations of motion are:...
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Schwarzschild's null-geodesic new form or an error?
My question is whether or not this form (radial acceleration of a photon)
$$\ddot{r}=\frac{L^2}{r^4}(r-3M)$$
is correct ?
Recall the standard set of second-order ODE for the Schwarzschild metric (for ...
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Test charge in 2+1 dimensions
Given a Chern-Simons theory,as in this resource(page 4), in 2+1 dimensions we can define the electric and magnetic fields as
$$
E_i=-\partial_iA_0-\partial_0A_i\;\;\;B=\epsilon^{ij}\partial_iA_j
$$
...
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Equations of Motion and Minimization of Spacetime Interval
I'm trying to show that the extrema of a path in spacetime, as given by the spacetime interval (or length if just considering space) is the one that solves the equations of motion.
Let a path be given ...
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Relation between equations of the form "Derivative" $f=0$
I'm currently taking an introductory course in QFT, and I've noticed that lots of equations in physics take the form of "Derivative" of a funcition equal 0. Some examples being the wave ...
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Euler-Lagrangian equation of motion of quantum fields in QFT
A canonical way of doing quantum field theory is by starting with some Lagrangian, for example, that of free scalar field
$$L=\frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi-\frac{1}{2}m\phi^2$$
Then ...
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What is the difference between 'Equation of motion' and 'Transport equation'?
I think this is a simple question with a not so easily explained answer. What is the difference between the Equation of motion and Transport eq? Is Navier stokes equation an 'Equation of motion' or a '...
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What the role of classical equation of motion in quantum field theory?
I've learnt quantum field theory for a semester but I still can't understand the role of classical equation of motion in QFT.
I have looked up for several books. They all discuss classical field ...
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Paradox in the Kinematic SUVAT Equations of Motion
The 5 equations of motion have been chosen such that from the 5 variables of motion: $s$, $u$, $v$, $a$ and $t$; each equation, exclusively omits one of these. This allows us to only ever require the ...
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Equations of motion of the metric in Classical Dilaton Gravity
In these lecture notes by Strominger section 3.3 we derive the equations of motion of the Classical Dilaton Gravity action
$$
S = \frac{1}{2\pi}\int{d^2x}{\sqrt{-g}e^{-2\phi}\left(R +4 (\nabla\phi)^2+...
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Heisenberg's equation of motion without reference to Schrödinger's picture
Well I'm reading Mukhanov & Winitzki's Introduction to quantum effects in gravity, and I got to the exercise 2.8 that ask to derive Heisenberg's equation of motion
\begin{equation}
\frac{d\hat{A}}{...
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Can energy conservation equation be seen as equation of motion?
After all, energy conservation equation is a differential equation that can be solved to find the motion, but this is never done. It is alway considered equation of motion only the time derivative of ...
user291161
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Why do we need to equation of state (EOS) equation?
I cannot understand why we need the equation of state of star for the numerical solution of TOV equation. In fact, I do not know how EOS equation helps with solving TOV equation. How is EOS equation ...
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Newton-Euler approach for the equations of motion of spherical pendulum attachted to a moving body with 6 degrees of freedom
I would like to obtain equations of motion for a spherical pendulum suspended from a (6 degrees of freedom) moving body (generalization of a simple pendulum on a cart) using Newton-Euler approach. ...
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Trajectory of magnetic dipole in a inhomogeneous magnetic field
Goal:
I want to use Python to illustrate how a magnetic dipole with magnetic moment m2 moves in a non-homogeneous magnetic field in a 2D-Plane. This field is generated by another magnetic dipole with ...
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How do I derive a particular form of the Langevin equation from Mori's generalized master equation?
Preliminary: I have a generalized Langevin equation for a set of relevant operators, written in Liouiville space as $\{|A_{v}\rangle \}$, and obtained via Mori's projection operator technique:
\begin{...
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Can there exist a motion which cannot be described as a function of time?
I know there exist systems for which we cannot solve its differential equation, But I was wondering if there could be a motion that cannot be represented as a differential equation with respect to ...
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Equality modulo equations of motion [closed]
What does Qmechanic mean by “equality modulo equations of motion” when talking about Lagrangian formulation/formalism and so on?
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Question of ball falling down.Difficulty in understanding the formula
A ball is thrown upward from the top of a tower 40m high. u = 10m/s.Find time for it reach AD.
g = $10m/s^2$.
Taking upwards direction as +ve and downward as -ve.
u = +10m/s.$g=-10m/s^2$.s=-40.
$-40 = ...
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"Equations of motions" and direction of maximal entropy
Say one has a system of statistical physics whose entropy is given as a function of one or multiple variables; for example as $S(x)$. An example of such a system could be a osmosis system, or a ...
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I want to know the difference in two methods I have solved [closed]
A thief is driving away on a straight road in jeep moving with a speed of 9 m/s. A police man chases him on a motorcycle moving at a speed of 10 m/s. If the instantaneous separation of the jeep from ...
user280997
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Can we determine the order of the equations of motion simply by looking at the action?
Naively, one would expect the EL equations arising from an action to contain derivatives (of the dynamics field) of an order that is twice the order of the highest-order derivative (of the dynamic ...
user87745
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Deriving conserved charges from the equations of motion
It is very well established how to derive conserved charges associated to the symmetries of Lagrangian using the Noether's theorem. Also in the Hamiltonian formulation, we know how to derive the ...
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Bullet piercing through block problem [closed]
A bullet moving with a velocity of 200cm/s penetrates a wooden block and comes to rest after traversing 4cm inside it . What velocity is needed for travelling distance of 9cm in same block.
My though ...
user277768
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Lagrangian of a charged particle in a magnetic field (specific problem)
I have to determine the Lagrangian and the angular velocity $\omega = \dot\theta$, in cylindrical coordinates $(r, \theta, z)$, of a electron with mass $m$ and charge $-e$, wich is experiencing a ...
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Deriving the field operators for Quantum Field theories
I always see the form of the field operators derived by, in the case of a scalar spin 0 particle, imposing the field commutation relations on the classical field solutions of the Klein Gordon equation ...
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What is the difference between "field equations" and "equations of motion"?
I come across the terms "equations of motion" and "field equations" all the time, but what is the difference? For example, general relativity is described in terms of the Einstein ...
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Can we reverse the geodesic equation to find a metric for the theory?
The geodesic equation describes the motion of a particle moving in a straight line embedded in a curved geometry.
$$\frac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\...
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Motion of charged particle in uniform magnetic field and a radially symmetric electric field
This question posted by me on MSE talks about a physics problem. This is what it was: (I hope someone can help me with this)
Consider a region of 2-dimensional space with a uniform magnetic field of ...
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Missing equations in Maxwell's equations
We have Maxwell's Equations (ignoring permittivity and permeability of free space)
$$
\nabla\cdot E=\rho\;;\;\nabla\times E=-\frac{\partial B}{\partial t}
$$
$$
\nabla\cdot B=0\;;\;\nabla\times B=\...
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Velocity of object on extremely low air density?
If we derive velocity in air when setting air resistance to $kv$,
we'll get
$$v= \frac{mg}{k}\left(1-e^{\frac{kt}{m}}\right) $$
and if air density goes to $0$, $k$ will also goes to $0$.
When $t=T$ (...
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Equations of motion describing a great circle
I'd like to argue that equations of motions of the form
$$\ddot \varphi = 0 \quad \text{and} \quad \ddot\theta = \sin\theta\cos\theta\dot\varphi^2$$
describe a great circle.
I think the standard ...
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numerically integrating a trajectory in polar coordinates
So I've reduced my problem to not being sure how to integrate a trajectory in polar coordinates. Suppose I have a free particle and I express its Hamiltonian thus:
$H =\eta_{ij}P^iP^j,$
where $\...
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Why can we use the equation of motion to calculate the amplitude in "Quantum Field Theory"?
I am reading the chapter on electron-proton scattering from "Quantum Field Theory in a Nutshell". The author calculates the amplitude of the electron-proton scattering (up to the second order). The ...
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In what sense are the equations of motion conserved by symmetries?
I am studying variational principles and I have been reading this set of notes by Townsend. In the first paragraph of Section 9, Townsend defines what it means for a transformation to be a symmetry of ...
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Rotation as an example of symmetry in classical mechanics
I modified the question because it was confused.
On my book there is this mathematical definition of symmetry transformation:
"The equations of motion have a symmetry, if the solutions of the ...
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Can we ignore the scalar field (dilaton) term in the Polyakov sigma-model action when deriving the classical equations of motion?
I have the full Polyakov sigma model action:
\begin{equation}
\begin{split}
&S=S_P + S_B + S_\Phi = \\
&- {1 \over 4 \pi \alpha'} \Big[ \int_\Sigma d^2\sigma \sqrt{-g} g^{ab} \partial_a X^\mu ...
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Why intuitively, do we define symmetries as transformations that map solutions of the equations of motion into other solutions?
Of course, strictly speaking, a symmetry is always a transformation that leaves a given object unchanged. But I'm curious why observable symmetries of physical systems are exactly those ...
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