What is Equations of motion: Definition and 234 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.
Hey so I encountered a problem recently, which while being simple, gave me another problem that I couldn't solve.
The problem:
An object is launched vertically upward with an initial velocity of 10 m/s from an elevation of 20m and allowed to hit the ground. Develop the equations of motion...
I want to compute the equations of motion for this theory in terms of the functions ##f## and ##a##. My plan was to apply the Euler-Lagrange equations, but it got confusing very quickly.
Am I right that we'll have 3 sets of equations? One for each of the fields ##\phi,\phi^\dagger, A_\mu## ...
The virtual displacement should be given by
$$
\delta\vec{r} = \begin{pmatrix} \cos(\alpha) \\ \sin(\alpha) \\ \end{pmatrix} \delta s
$$
where ##\delta s## is a displacement parallel to the plane. The relevant force should be the gravitational force, as given above. Thus, the equations of...
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
$$i \gamma^{\mu} \partial_{\mu} \psi = m \psi_c \\
i \gamma^{\mu} \partial_{\mu} \psi_c = m \psi
$$
Where ##\psi_c = C \gamma^0 \psi^*##
Show that the above equations can be obtained from the followong lagrangian
$$
L = \overline{\psi} i \gamma^{\mu} \partial_{\mu} \psi - \frac{1}{2} m \left...
Good morning, I'm not a student but I'm curious about physics.
I would like to calculate the equation of motion of a system using the Lagrangian mechanics. Suppose a particle subjected to some external forces.
From Wikipedia, I found two method:
1. using kinetic energy and generalized forces...
The Lagrangian for a massless particle in a potential, using the ##(-,+,+,+)## metric signature, is
$$L = \frac{\dot{x}_\mu \dot{x}^\mu}{2e} - V,$$
where ##\dot{x}^\mu := \frac{dx^\mu}{d\lambda}## is the velocity, ##\lambda## is some worldline parameter, ##e## is the auxiliary einbein and...
In the holonomic case, we can put the Lagrangian in the Lagrange equations to obtain the explicit form of the equations of motion. From Greenwood's classical dynamics book, the equations are
Are there such general equations for the non-holonomic case?
I'm working through some things with general relativity, and am trying to solve for my equations of motion from the Schwarzschild Metric. I'm new to nonlinear pde, so am not really sure what things to try. I have 2 out of my 3 equations, for t and r (theta taken to be constant). At first glance...
Greetings Good People,
As the title suggests, I'm having some trouble getting to a 2D model. The process is to select an aircraft (or wing model), and model it as a 2D, 2DOF wing-tunnel model.
The aircraft I selected was a Cessna 172. This had a tapered wing, which after some calculations and...
Take rightwards as positive.
There are 2 equations of motion, depending on whether ##\frac {dx} {dt} ## is positive or not.
The 2 equations are:
##m\ddot x = -kx \pm \mu mg##
My questions about this system:
Is this SHM?
Possible method to solve for equation of motion:
- Solve the 2nd ODE...
Summary:: I have come across a situation where I seem to get different equations of motion for an oscillating system. Please do help me find out where I went wrong.
*I am not asking how to solve the problem*
I am going to consider 4 parts of the cylinder's motion, as listed below. (There is...
Summary:: What are the Equations of motion for a free damped 2-Dof systrem?
Hello,
I am required to calculate the equations of motion for a 2-dof system as shown in the attached file. The system is undergoing free damped vibrations. I have found the equations of motion for no damping but i...
So what I did was made the two equations equal each other. A lot of stuff cancels out and I end up with x=-vt. My issue is that t isn't given and I am not entirely sure how to get it. I don't think taking the partitial derivative of time will be any help nor the partial derivative of...
Hello,
I'm doing an FBD exercise and I'm hoping you guys and girls can give me some feedback. I've already done a lot, but I'm not completely sure about it and since it's part of my Msc Thesis, it's important that I get it right. After setting up the FBDs, I get 5 equations and 6 unknown...
I need help to understand this problem taken from Mechanical Vibrations by S. Rao
I know that the equations of motion could be obtained in various ways, for example using the Lagrangian, but, at the moment, I am interested in understanding the method he used. In particular, if I'm not...
In the Schwarzschild metric case why is : $$r=cste,$$
$$\theta(t)=\arccos(\sin\alpha\cos\omega t)),$$
$$\phi(t)=\arctan(\tan(\omega t)/\cos\alpha),$$
$$t=\tau$$
not solution to the geodetic equation ?
Hello,
I have a system with 2 degrees of freedom with 2 non-holonomic constrains that can be expressed by:##A_1 dq_1 +Cdq_3 + Ddq_4 = 0##
##A_2 dq_1 + Bdq_2 = 0##Being ##q_1, q_2, q_3## and ##q_4## four generalized coordinates that can describe the movement of the system. And ##A_1, A_2, B...
This problem fascinated me in lower division physics. Find the 2-d equations of motion for a Slinky going down a flight of stairs (assuming the path of the slinky is planar; eg only going up and down and front and back, no side to side). I do confess that whilst I do love physics I’m not...
I think that the only force acting on the wall is the normal force caused by Coriolis force, so it can be calculated this way:
##N=m2\dot r \dot \theta##
But ##\dot r## is not constant, so how can I calculate it?
Then, I can't calculate the acceleration either since I don't have the value of...
In the picture, there is a problem where the t is in units of square root(l/g), and V in square root(gl)
I am wondering
1. What it means when time is in units other than time? Does it mean that when solving I have to take time/squareroot(l/g)
2. How did they get square root(l/g).
Thank you...
To begin with, I posted this thread ahead of time simply because I thought it may provide me some insight on how to solve for another problem that I have previously posted here: https://www.physicsforums.com/threads/special-relativity-test-particle-inside-suns-gravitational-field.983171/unread...
Summary:: This is a system and we want to find the equations of motion. After some force-based attempts, I think that it would be easier to use some energy methods.
Hi,
I wanted to ask about deriving equations of motion by using the Lagrangian. The question is in the picture below. We are...
Hi!
First of all, mention that this is not a "homework" problem in the sense that no teacher ever gave it to me or that I have the obligation to do it. It is a question that came to mind when repasing the theory done in class and though interesting. I still post it here because I suppose that...
I'm stack at the very beginning. If I use Newton's second law to find acceleration and integrate until I find the position, I must face
$$v(t) = \int_0^t g-\frac{\lambda v}{m} dt'=gt-\frac{\lambda }{m} \int_0^t\frac{\partial z}{\partial t}dt$$
But this last term feels pretty weird. I don't...
The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative. Also I know for a fact that my Lagrangian is correct.
I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and...
I have attached the problem and I have also taken a photo of my working for it, it seems my answer is wrong.
I would be grateful if you could please check where I have gone wrong.
Thanks a lot.
First of all, disclaimer: This isn't an official assignment or anything, so I'm not even sure if there is a resonably simple solution.
Consider the following sketch.
(Forgive me if it isn't completely clear, I didn't want to fiddle around for too long with tikz...)
Let us assume that we can...
Here is a picture of the problem.
I have chosen the origin to lie in the middle of the circle around which the mass moves. I have also chosen the z axis to pass through the origin and through the vertex of the right circular cone. The x-axis and y-axis are so that one when curls his or her...
Homework Statement
I'd like to derive the equations of motion for a system with Lagrange density
$$\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...
I would like to preface this by saying that I solved the homework problem, but my professor gave me an added challenge of finding the period of the motion described in this problem.
1. Homework Statement
The pendulum bob of mass m shown in the figure below is suspended by an in-extensible...
Homework Statement
The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system).
I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm...
Homework Statement
We have been given the following mass-spring-damper diagram, and are asked to derive the equations of motion. The positions of the two masses are given as q1 and q2.
The Attempt at a Solution
I began by drawing free-body diagrams for each mass.
Then I set up the...
<<Moderator's note: Moved from a technical forum, no template.>>
Description of the system:
The masses m1 and m2 lie on a smooth surface. The masses are attached with a spring of non stretched length l0 and spring constant k. A constant force F is being applied to m2.
My coordinates:
Left of...
Say, I have a system at rest. I was wondering - how many equations of motion can the system have (without redundancy)? Well, I thought that equating the forces along 2 or 3 different axes would give 3 independent equations. Also equating torques would give some equations, but how many of them...
Homework Statement
This is an example problem I found on khan academy and it didn't have an official problem statement... So I am going to have to make up my own problem statement from what was given. I can link the video if any of you want to see it.
A positive charge 4uC and a negative...
A massless spin 1 particle has 2 degrees of freedom. However, we usually describe it using four-vectors, which have four components. Hence, somehow we must get rid of the superfluous degrees of freedom. This job is done by the Maxwell equations. To quote from Gilmore's "Lie Groups, Physics, and...
Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
I made the problem up myself, so there might very well not be a rational answer that I like!
Homework Statement
A point-particle is released at height h0 is released into a parabola. The position of the particle is given by (x, y) and the acceleration due to gravity is g. All forms of friction...
Suppose one starts with the standard Klein-Gordon (KG) Lagrangian for a free scalar field: $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}$$ Integrating by parts one can obtain an equivalent (i.e. gives the same equations of motion) Lagrangian...
Homework Statement
Homework Equations
The equations are all given
The Attempt at a Solution
This isn't really a homework question... it's solved, but I'm having a hard time following it. I don't understand where the first term:
-a(kaθ)
comes from. I can see it's the moment relation...
Hello there, I've been considering the geodesic equations of motion for a test particle in Schwarzschild geometry for some time now. Similar to what we can do with the Kepler problem I would like to be able to numerically integrate the equations of motion. I'm quite interested to see how...
Hello! I've been looking into the equations of motions of a quadcopter with an x-config. The only thing I've found was http://www.inase.org/library/2014/santorini/bypaper/SYSTEMS/SYSTEMS1-07.pdf, but it seems complicated. Could someone explain the equation of motions, and how they were arrived...
Homework Statement
Thin uniform disk with radius r, mass m, and moment of inertia 0.5mr2 is suspended from a cable line where one end is attached to a set point via a spring, and the other end is also attached to a spring but is moving in an upwards direction. Solve for the equations of motion...
I have seen many examples of the EOM for a rocket derived for the following cases:
No gravity, No drag
Gravity, No drag
No gravity, linear drag (b*v where b is a constant)
I have never seen
Gravity, linear drag
Gravity, quadratic drag
I took John Taylor's two examples of linear drag and...