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Questions tagged [equations-of-motion]

DO NOT USE THIS TAG just because the question contains an equation of motion!

DO NOT USE THIS TAG just because the question contains an equation of motion!

Almost all physical systems have equations of motion, with the possible exception of static systems. Hence it is often a poor way to classify a question to use this tag.

81 questions
109
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12 answers

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask me that question, then I'd probably mumble something…
Nikolaj-K
  • 8,873
43
votes
7 answers

Is there a proof from the first principle that the Lagrangian $L = T - V$?

Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$…
Chin Yeh
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23
votes
3 answers

What does a Lagrangian of the form $L=m^2\dot x^4 +U(x)\dot x^2 -W(x)$ represent?

I saw this Lagrangian in notes I have printed: $$ L\left(x,\frac{dx}{dt}\right) = \frac{m^2}{12}\left(\frac{dx}{dt}\right)^4 + m\left(\frac{dx}{dt}\right)^2 V(x) -V^2(x). $$ [It appears in Chapter 1 of Goldstein's Classical Mechanics as Exercise 18…
0x90
  • 3,456
13
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3 answers

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 transformations that map solutions of the equation of motion…
jak
  • 10,431
10
votes
3 answers

Is there any case in physics where the equations of motion depend on high time derivatives of the position?

For example if the force on a particle is of the form $ \mathbf F = \mathbf F(\mathbf r, \dot{\mathbf r}, \ddot{\mathbf r}, \dddot{\mathbf r}) $, then the equation of motion would be a third order differential equation, what will require us to know…
Andrey S
  • 1,066
9
votes
5 answers

Does the stress-energy tensor contain the equations of motion?

Derivatives $\nabla_i T^{ik}=0$ of a stress-energy tensor of physical system express conservation laws. Whether contains a stress-energy tensor also the information on the equations of motion of system?
Sergio
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8
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3 answers

What is the relationship between Schrödinger equation and Boltzmann equation?

The Schrödinger equation in its variants for many particle systems gives the full time evolution of the system. Likewise, the Boltzmann equation is often the starting point in classical gas dynamics. What is the relationship, i.e. the classical…
Nikolaj-K
  • 8,873
7
votes
1 answer

Can I really take the classical field equations at face value in QFT?

To be concrete, let's say I have a relativistic $\phi^4$ theory [with Minkowski signature $(+,-,-,-)$] $$ \tag{1} \mathcal{L} ~=~ \frac{1}{2} \left ( \partial_{\mu} \phi \partial^{\mu} \phi - m^2 \phi^2\right ) - \frac{\lambda}{4!} \phi^4. $$ The…
Pth
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7
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3 answers

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 by employing the Euler-Lagrangian equation, i.e.…
Tan Tixuan
  • 460
6
votes
4 answers

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 field equation $G_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu…
Superbee
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6
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2 answers

Question about Type IIB supergravity equations of motion

This is probably a dumb question, but I'm a mathematician who's been trying to understand the equations of motion for Type IIB supergravity, and I'm not quite sure I understand what's going on with the Einstein equations. Specifically, I'm following…
Mark B
  • 751
5
votes
2 answers

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 change in the Lagrangian corresponding to an…
irmbil
  • 95
5
votes
2 answers

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 theory. And they turn to the quantum part without…
Taveren Sa
  • 189
5
votes
2 answers

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. $$\frac{d^2 x^\mu}{d\tau^2}+\Gamma^\mu_{\alpha\beta} \frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau}=0$$ Solving for the proper…
Joshua Pasa
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5
votes
3 answers

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 a system: We have $$I[\mathbf…
MB10000
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