Questions tagged [equations-of-motion]

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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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187 views

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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30 views

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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36 views

Does the equation of ball going up in gravity straight up with a vertical initial velocity hold when it's coming down?

Suppose a ball is thrown up and having the same altitude at different times. So should the equation of motion $ v_0t - gt^2$ which I derive from balancing forces hold for both the times when the ball ...
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208 views

Upward Moving Cable Car [closed]

A $1500$ kg cable car moves vertically by means of a cable that connects the ground and the top of the hill. What is the tension in the supporting (massless) cable when the cab, originally moving ...
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2answers
125 views

Can an Equation of Motion Do More?

My usual expectation is that an equation of motion should give me the time-evolution of a system given an initial condition. But I am curious as to can an equation of motion do more than that? In ...
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120 views

Are symmetries in the equation necessarily symmetries in the corresponding solution(s)?

I wonder whether the symmetries in the equations (such as the heat equation, the wave equation, the Schrödinger equation, Maxwell equations) are reflected into their solution(s). I.e., assuming that ...
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187 views

Equations of motion of a cylinder on a horizontal plane

How would I go about deriving the equations of motion for the motion of the centre of mass of the cylinder in this system: The cylinder has mass $M$ and radius $R$ and the small mass $m$ is being ...
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1answer
465 views

Transformation of generalized coordinates

One of the advantages of Lagrangian formulation is that the equations of motion have the same form regardless of the choice of generalized coordinates. Suppose that a system has $s$ degrees of freedom,...
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133 views

Do the equations of motion have specific characteristics?

I solved a classical mechanics problem in a form somewhat like this: $$x(t)=t^2+5t$$ $$y(t)=t^3$$ $$z(t)=5.$$ The problem asked me to find the equations of motion of an object. From my ...
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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 ...
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94 views

What does it mean to find an equation of motion, given vector functions that describe both the object's position and velocity?

I don't really understand how to approach a problem that asks to find the equation of motion. Intuitively, I would guess that an "equation of motion" is an equation where the particle's position is ...
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203 views

Showing the invariance of the equations of motion

It is strange to me that for a symmetry which involves $\dot{x}$, there seems to always appear a term with $\dddot{x}$ in the variation of the equations of motion, which doesn't makes much sense. I ...
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Is it valid to replace the equations of motion inside a symmetry?

For example, this symmetry: $$\delta q^{i}=\epsilon(q^{i}-2\dot{q}^{i}t)$$ it's derivative is: $$\delta\dot{q}^{i}=-\epsilon(\dot{q}^i +2\ddot{q}^i t)$$ There appears $\ddot{q}^{i}$ in this ...
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202 views

Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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88 views

Physical significance of omitting a purely time dependent term from a Lagrangian

For a simple pendulum whose point of support moves on a vertical circle of radius $a$ with constant frequency $\gamma$, you can write the Lagrangian down. The potential energy can be written as $-mg(-...
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321 views

Why isn't the Time-Independent Schrödinger Equation an equation of motion?

I thought an equation of motion was something where you are given a Lagrangian and, using the Euler-Lagrange equation, you then find the equations of motion for that system. Same basic idea for the ...
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What is a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$?

As I understand it, the Euler-Lagrange equation is a necessary but not a sufficient condition to determine if the action integral reaches an extremal through the trajectory described by $q(t)$. If ...
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707 views

What is an “equation of motion” as used in context of geodesic equation?

I am studying general relativity and using the book Gravity by James Hartle. On page 170, he provides the following table: I don't understand what he means by "equation of motion" nor do I ...
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What is “first order“ and “second order” in time?

What is the meaning of the text quoted below? In the physical world, if a system is described by an equation that is first order in time, the system is general dissipative (has energy loss). If ...
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165 views

Does spatial coupling prohibit resonances due to an external source field?

The harmonic oscillator coupled to a sinodial external source $$\frac{\partial^2 x(t)}{\partial t^2}+\omega_0^2 x(t)=F_0\sin(\omega_\text{ext}\ t),$$ has the solution $$x(t)=x(0)\cos(\omega_0 t)+C \...
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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 ...
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General equation of motion for elementary particles

Elementary particles can be grouped into spin-classes and described by specific equations, see below: Is there a general Lagrangian density from which all these equations can be derived? A ...
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653 views

Control system with equation C = A*x + B*dx/dt

This question came up in a computer science / robotics exam and I still don't know the solution for it. I figured out that it's classical mechanics related, so I thought this might be the best place ...
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Can a force in an explicitly time dependent classical system be conservative?

If I consider equations of motion derived from the principle of least action for an explicitly time dependent Lagrangian $$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$ under what ...
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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. ...
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724 views

Laws of Motion: Acceleration to be applied on a free falling object to reduce velocity to 0

Assume that an object of mass 'm' is falling to the earth.The force acting on the same would be F = m*g = 10m (assuming g = 10m/s^2). In this case the velocity of the object at time 't' = 10 seconds ...
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290 views

Equation $H(q,p)=E$ is the equation of motion or energy-conservation law?

I do not completely understand, why do we consider Hamilton–Jacobi equation $H(q,p)=E$ as equation of motion, whereas it is looks like energy-conservation law?
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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 ...