What is Equation of motion: Definition and 266 Discussions
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.
There are two main descriptions of motion: dynamics and kinematics. Dynamics is general, since the momenta, forces and energy of the particles are taken into account. In this instance, sometimes the term dynamics refers to the differential equations that the system satisfies (e.g., Newton's second law or Euler–Lagrange equations), and sometimes to the solutions to those equations.
However, kinematics is simpler. It concerns only variables derived from the positions of objects and time. In circumstances of constant acceleration, these simpler equations of motion are usually referred to as the SUVAT equations, arising from the definitions of kinematic quantities: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t).
Equations of motion can therefore be grouped under these main classifiers of motion. In all cases, the main types of motion are translations, rotations, oscillations, or any combinations of these.
A differential equation of motion, usually identified as some physical law and applying definitions of physical quantities, is used to set up an equation for the problem. Solving the differential equation will lead to a general solution with arbitrary constants, the arbitrariness corresponding to a family of solutions. A particular solution can be obtained by setting the initial values, which fixes the values of the constants.
To state this formally, in general an equation of motion M is a function of the position r of the object, its velocity (the first time derivative of r, v = dr/dt), and its acceleration (the second derivative of r, a = d2r/dt2), and time t. Euclidean vectors in 3D are denoted throughout in bold. This is equivalent to saying an equation of motion in r is a second-order ordinary differential equation (ODE) in r,
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{\displaystyle M\left[\mathbf {r} (t),\mathbf {\dot {r}} (t),\mathbf {\ddot {r}} (t),t\right]=0\,,}
where t is time, and each overdot denotes one time derivative. The initial conditions are given by the constant values at t = 0,
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{\displaystyle \mathbf {r} (0)\,,\quad \mathbf {\dot {r}} (0)\,.}
The solution r(t) to the equation of motion, with specified initial values, describes the system for all times t after t = 0. Other dynamical variables like the momentum p of the object, or quantities derived from r and p like angular momentum, can be used in place of r as the quantity to solve for from some equation of motion, although the position of the object at time t is by far the most sought-after quantity.
Sometimes, the equation will be linear and is more likely to be exactly solvable. In general, the equation will be non-linear, and cannot be solved exactly so a variety of approximations must be used. The solutions to nonlinear equations may show chaotic behavior depending on how sensitive the system is to the initial conditions.
Well, I started with the first equation of motion for the scalar field, but I'm really not sure if I'm doing it the right way.
\begin{equation}
\begin{split}
\frac{\partial \mathcal{L}}{\partial \varphi} &= \frac{\partial}{\partial \varphi} [(\partial_\mu \varphi^* -...
Problem statement : I copy and paste the problem as it appears in the text down below. I have only changed the symbol of the given acceleration from ##a\rightarrow a_0##, owing to its constancy.
Attempt : I must admit that I could proceed very little.
Given...
I am reading the following notes: https://arxiv.org/pdf/hep-th/9902115.pdf and am trying to make the connection between equations (22) and (24). Specifically, I do not understand how they were able to get (24) from (22) using the dual field prescription. I guess naively I'm not even sure where...
The rocket has constant velocity so we can write it's equation of motion as :
$$\vec r = \vec r_0+t \vec v_0 $$ We can write it for components along each axis :
$$x = x_0 + v_{0,x}t$$$$y = y_0 + v_{0,y}t$$$$z = z_0 + v_{0,z}t$$We put known values in above equations...
For this,
I don't understand why they don't have a negative sign as the torque to the friction should be negative. To my understanding, I think the equation 5.27 should be ##I\frac{d \omega}{dt} = -F_{friction}R## from the right hand rule assuming out of the page is positive.
Noting that ##f_k...
We have ##U(x) = U(a) + \frac{1}{2}U''(a)(x-a)^2## (by taylor series)
It's known that ##U'(a) = 0## and point ##a## is a turning point, hence at that point, Kinetic energy is 0, and ##E = U(a)##, hence:
We have ##U(x) = E + \frac{1}{2}U''(a)(x-a)^2##I need to get equation of motion and I want...
Hello! I have a Lagrangian of the form:
$$L = \frac{mv^2}{2}+f(v)v$$
where ##f(v)## is a function of the velocity. I would like to derive the equation of motion in general, without writing down an expression for ##f(v)## yet. I have that ##\frac{\partial L}{\partial x} = 0##. However, what is...
I want to use the Lagrangian approach to find the equation of motion for a mass sliding down a frictionless inclined plane. I call the length of the incline a and the angle that the incline makes with the horizontal b. Then the mass has kinetic energy 1/2m(da/dt)2 and the potential energy should...
Hi guys,
I can't seem to be able to get to
$$ (\rho + p) \frac {d\Phi} {dr} = - \frac {dp} {dr} $$
from
$$T^{\alpha\beta}_{\,\,\,\,;\beta} = 0$$
the only one of these 4 equations (in the case of a spherically symmetric static star) that does not identically vanish is that for ##\alpha=r##...
Hello Everyone
I want to model forces affecting on syringe plunger , but I do not know how to calculate terms like friction and damping coefficient.
What I imagine is that : F_driving = ma + cv + f ----------------(1)
where:
f: friction
c: coefficient of viscous damping
m: mass of plunger (is...
The system is shown below. It consists of a rod of length ##L## and mass ##m_b## connecting a disk of radius ##R## and mass ##m_d## to a collar of mass ##m_c## which is in turn free to slide without friction on a vertical and rigid pole. The disk rolls without slipping on the floor. The ends...
I am looking at a textbook solution to the following problem of finding the equation of motion of a half disk. In the solution, the author considers the half disk has a COM at the black dot, and to find the instantaneous translational velocity of the center of mass (he considers rotational...
Good morning, I'm not a student but I'm curious about physics.
I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces.
From Wikipedia, I found two method:
1. using kinetic energy and generalized forces...
Summary:: Differential equation of motion, parabola
Hi. I've tried resolve this problem but I have two doubts. The first is about the differential equation of motion because I can't simplify it to the form y" + a*y' + b*y = F(t). I'm not sure if what I got is right. My second doubt is that I...
I'm surprised that this question only occurred to me recently. If a have an electrically charged mass attached to a spring and set it oscillating, the resulting production of electromagnetic waves must cause a kind of "friction", a force resisting the motion of the charged mass, so that its...
I was wondering how the second equation of motion produces negative displacements ##s= ut+\frac 12 at^2## . Is ##\frac12 at^2## kind of distance operator?
When I use Lagrange to get the equations of motion, in order to find the equilibrium conditions I set the parameters q as constants thus the derivatives to be zero and then calculate the q's that satisfy the equations of motion obtained.
In ordert to check about stability I think I need to add...
I think I undeerstand Lagrangian mechanics but I have a question that will help to clarify some concepts.
Imagine I throw a pencil. For that I have 5 generalised coordinates (x,y,z and 2 rotational).
When I express Kinetic Energy (T) as:
$$T = 1/2m\dot{x^{2}}+1/2m\dot{y^{2}}+1/2m\dot{z^{2}} +...
My understanding of the system from the image (which was given in book)
I could see there's 3 tension in 2 body. Even I had seen 2 tension in a body. It was little bit confusing to me. I could find tension in Lagrangian from right side. But left side was confusing to me...
It's the body. So there's friction on that plane and there's tension also.
$$L=\frac{1}{2}m_1\dot{x}^2+\frac{1}{2}m_2\dot{x}^2-m_2g(l-x)-m_1gx\sin\theta$$
$$f=\mu N=-\mu m_1 g\dot{x}\cos\theta$$
I had found the frictional force's equation from [the...
Hello,
Given the statement a described above. To find the forces at point D I drawn a kinematic scheme and FBD of rod CD. But why am I allowed to ignore the mass of 50 kg, the forces at point B and point A? I know the are some rules about this, but I just can't remember them anymore.. The...
Hi guys,
I have started studying differential equations on my own, sticking to my last high-school year's textbook, along with a few physics applications of ODEs.
Online I came across the n-body problem, which then took me to the basic two-body problem!
I'm here to ask you a few things about...
Heisenberg equation of motion for operators are given by
i\hbar\frac{d\hat{A}}{dt}=i\hbar\frac{\partial \hat{A}}{\partial t}+[\hat{A},\hat{H}].
Almost always ##\frac{\partial \hat{A}}{\partial t}=0##. When that is not the case?
In the figure assume the "ceiling" moves with motion ##Y(t)##, i.e. it is a point support.
Applying Newton's law in the vertical direction
##T(y).\hat{y}=\rho y[g+\frac{d^{2}Y}{dt^{2}}]##
If ##\theta## is the angle between ##T## and ##\hat{y}## that means ##|T|\cos\theta=\rho...
Hi,
I have a particle on a parabolic surface $$y = Ax^2$$ and I have to show that the frequency is $$\omega = \sqrt{2Ag}$$
I don't know how to deal with a parabola. I don't think I can use the polar coordinates like a circle.
I don't see how to start this problem and in which coordinates...
Hello! I've done the following to solve this problem.
a) Here I simply put in the time in the equation, s0 is = 0 and after that it was pretty much done
$$s(t) = 42 *1 - \frac {9,81*1^2} {2} = 37,09m $$
b) Now here to see when the rocket reaches it maximum altitude and what height it is...
Hello to everyone :smile:
I'd like to study this problem.
You have a 2D guide, described by an equation y = y (x) in a reference interval x ∈ I = [a, b], placed in a cartesian vertical plane Oxy.
The guide is frictionless and the only force that is acting is the gravity force.
On this track, a...
Consider the classical Heisenberg model without an external field which is defined by the Hamiltonian:
\mathcal{H} = -\sum_{ij} J_{ij} \vec{s_i}\vec{s_j}
where J_{ij} > 0 describes the coupling between the spins \vec{s}_i \in \mathbb{R}^3 on some lattice. (Is there a way to use tex...
1) We know that for a given Killing vector ##K^\mu## the quantity ##g_{\mu\nu}K^\mu \dot q^\nu## is conserved along the geodesic ##q^k##, ##k\in\{t,r,x,y\}## . Therefore we find, with the three given Killing vectors ##\delta^t_0, \delta^x_0## and ##\delta^y_0## the conserved quantities
$$Q^t :=...
This is a question from an exercise I don't have the answers to.
I have been trying to figure this out for a long time and don't know what to do after writing
mx''¨(t)=−kx(t)+mg
I figure that the frequency ω=√(k/m) since the mg term is constant and the kx term is the only term that changes.
I...
Completely new to this and wondered if someone can explain what the correct equation of motion is if x is extension, t is time and a,b are constants
x = b log (at)
x = a t exp(-b t)
x = exp(-b t) sin(a t)
x = a sin(t)/b
a)
Alright here we have to use Euler-Lagrange equation
$$\partial_{\alpha} \Big( \frac{\partial \mathcal{L}}{\partial(\partial_{\mu} A_{\nu})} \Big) - \frac{\partial \mathcal{L}}{\partial A_{\nu}} = 0$$
Let's focus on the term ##\frac{\partial \mathcal{L}}{\partial (\partial_{\alpha}...
The equation of motion of a simple pendulum is:
$$\ddot \theta + \frac{g}{l} \theta = 0$$
Our Physics professor told us: 'If you want to become a good Physicist you have to be able to analytically check your answers to see whether they make sense'.
In class he took the limits of constant...
Exercise statement:
Given the action (note ##G_{ab}## is a symmetric matrix, i.e. ##G_{ba} = G_{ab}##):
$$S = \int dt \Big( \sum_{ab} G_{ab} \dot q^a\dot q^b-V(q)\Big)$$
Show (using Euler Lagrange's equation) that the following equation holds:
$$\ddot q^d +...
Homework Statement: finding equation of motion for Born-Infeld lagrangian
Homework Equations: born-infelf lagrangian
i do not know where I'm going wrong.
i'll be really grateful for any advice.
A solution of equations of motion for charged particle in a uniform magnetic field are well known (##r = const##, ## \dot{\phi} = const##). But if I tring to solve this equation using only mathematical background (without physical reasoning) I can't do this due to entaglements of variables...
So ##T+U=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2})-mgy=constant##. If I derive this with respect to ##t##
$$\dot{x}\ddot{x}+\dot{y}\ddot{y}-g\dot{y}=0$$
Then I use ##\dot{y}=\dot{x}\frac{dy}{dx},\ddot{y}=\ddot{x}\frac{dy}{dx}+\dot{x}^{2}\frac{d^{2}y}{dx^{2}}##
to get...
Homework Statement
A) By examining the effective potential energy find the radius at which a planet with angular momentum L can orbit the sun in a circular orbit with fixed r (I have done this already)
B) Show that the orbit is stable in the sense that a small radial nudge will cause only...
<< Mentor Note -- thread moved from the technical forums, so no Template is shown >>
Show, from the first principles, that the equation of motion of a mass (m) on a spring, subjected to a linear resistance force R, a restoring force S, and a driving force G(t) is given by
d2x/dt2+ 2K(dx/dt) +...
Homework Statement
I'd like to derive the equations of motion for a system with Lagrange density
$$\mathcal{L}= \frac{1}{2}\partial_\mu\phi\partial^\mu\phi,$$
for ##\phi:\mathcal{M}\to \mathbb{R}## a real scalar field.
Homework Equations
$$\frac{\partial...
Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion:
Mg*sin{α} - 1.5m*x(double dot)=0
I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...
Homework Statement
equations above are descriptive of a system with two configuration variables, q1 and q2. inputs are tau1 and tau2. d and c values are given.
the question is about conversion of above equations to a state-space equation where the state-variables are x1 = q1_dot, x2 = q1_2dot...
Homework Statement
The problem is taken from Morin's book on classical mechanics. I found out Lagrangian of motion. Now to solve, we need small angle and small x approximation. The small angle approximation is easy to treat. But how to solve small x approximation i.e how do I apply it...
Hi
There is an example in my textbook worded as follows;
A particle of mass 2kg moves along the positive x-axis under the action of a force directed towards the origin. At time t seconds, the displacement of P from O is x metres and P is moving away from O with a speed of v ms^-1. The force has...
Homework Statement
The evolution of the density in a system of attractive spheres can be described by the following dynamic equation.
$$\frac{\partial}{\partial t} \rho (r,t) = D_o [\nabla^2 \rho (r,t) + \beta \nabla \rho (r,t) \int dr' [\nabla V (|r-r'|)] \rho (r',t) g(r,r',t)]$$
a)...