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.
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.
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I have a particle on a parabolic surface $$y = Ax^2$$ and I have to show that the frequency is $$\omega = \sqrt{2Ag}$$
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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.
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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:
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Homework Statement
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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...
So if you have a rocket let's say that discards all the structural and engine mass continuously at zero velocity that is relative to the rocket until only the payload is traveling at the final velocity - then what will the equation of motion will look like? we can neglect the drag and...
Homework Statement
This problem showed up in my final review packet, and I /think/ it should be basic kinematics, but I don't even know how to approach it with the second half of it.
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Homework Statement
I am not looking for a solution to the problem, as much as I need a clarification on what it's asking for. The problem:
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Martyna
Thread
drag force
equationofmotion
inclined plane
mechanics
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Homework Statement
Andy Roberts the former West Indian Cricket player and Fast Bowler, bowled his fastest delivery in 1975 at 159.5 km/h. Neglecting air resistance, calculate the maximum distance he could have thrown the ball at this speed (on earth!) had he been able to throw it:
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Homework Statement
I think I made a mistake somewhere..
Homework Equations
T = Jα
T = F*R
The Attempt at a Solution
A)
I started with T = Jα
Since there is no slip, αm = αL
Thus:
Tm / Jm = TL / JL
Plugging in, we find TL = Tm * JL / Jm = 2560
Now use T = F*R.
Tm = Fm * Rm
Plugging in...
A flying object is moving in 3D space having translational velocity and the object is also rotating. Consider a body frame (xb-yb-zb) attached to the C.G of the moving body. Hence the body attached frame is also translating and rotating (as the object is flying) with respect to a fixed inertial...
Homework Statement
A pendulum with a mass m hanging on a elastic bug rigid massless rod which may swing in the xy-plane. The pivot point is the origin of the coordinate system. The force acting on the pendulum is the sum of force of an elastic central force directed towards the origin, and...
Hi, i am an student of civil engineering and i am doing my graduate thesis. so sorry if i misspell, english is not my native languaje. so, here we go :D
Homework Statement
Before entering in details you (whoever you are) most know:
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Consider the 1d motion of a body under the influence of the force given by F = -m*γ*vα. m is mass, γ is a constant of appropriate dimension, v is velocity and α is dimensionless constant. The value of α for which the motion will come to a stop in finite time is to be calculated. I solved the...
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Hi all,
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Homework Statement
A projectile is launched horizontally with an initial velocity v0 from a height h. If it is assumed that there is no air resistance, which of the following expressions represents the vertical trajectory of the projectile? (A) h–gv0^2/2x^2 (B) h–gv0^2x^2 (C) h-gx^2/2v0^2 (D)...
Homework Statement
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Homework Statement
Homework Equations
3. The Attempt at a Solution [/B]
Hello guys,I posted images since its easier to write equations.Please can someone help me check this, if this is correct so far, then i should be able to find the velocity at C, using kinetic energy?
1. Problem Statement
Assume there is an rigid object with mass m in 2D space, an impulse J = FΔt is applied at time t1 at the particle Pimp and Pimp is on the exterior boundary of the object. The impulse cause a free plane motion of the object and the object is only affected by the force of the...
Hello Physics Forums!
Supposing that we have an action that says:
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Homework Statement
what is equation of motion for pendulum?pendulum is made of pointmass, which mass is m, is fixed to thread ,which length is l? Oscillation aplitude is θ.Other side of thread is fixed in (0;0;0)point. at time t=0
t=0;y=0
;z=0
;φ=θ
Homework Equations...
I have a problem (this is not homework)
Based on covariant Lagrangian ## \mathcal {L} = \frac {m}{2} \frac{dx^{\mu}}{ds} \frac {dx _ {\mu}}{ds} ## record the equations of motion in Hamiltonian form for a particle in the Schwarzschild metric (SM).
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