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Problem in deducing the equations of motion using indefinite integral 1 Kinematics: Given the velocity as a function of the position, is it possible to derive the velocity as a function of time?
The requirement is that y ″ = <const>. The reason these equations are called the kinematics equations is that they represent relationships between motion quantities without consideration of actual forces. All of the above expressions are a consequence of a single kinematics expression of the form y ″ = <const>.
"If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics." And "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.
The equations of motion for the field theory of general relativity are the Einstein equations. The equation of motion for a freely falling particle within general relativity is the geodesic equation. Both answers are correct, depending on what exactly you mean.
So we obtain the equation of motion for a particle moving in a metric space gμν, which is simply the geodesic equation (here I write it assuming λ is an affine parameter) : d2˜xμ d2λ + Γμρνd˜xρ dλ d˜xν dλ = 0. Where Γμνρ are the Christoffel symbols for the metric. Now, we must do the variation w.r.t. gμν(x), the metric.
The velocity, acceleration and position vectors are defined in terms of each other as follows: →v = d→x dt. →a = d→v dt. Using the previous two, you can obtain the third differential equation: →a = →vd→v d→x. We can rearrange the equations to obtain the following: d→x = →v dt. d→v = →a dt.
In the case of constant acceleration this gives the famous equation of uniformly accelerated motion: ax = 12v2 + C a x = 1 2 v 2 + C. Now, assume that at x = 0 x = 0 v(0) = u v (0) = u, where u is the initial speed so that C = −12u2 C = − 1 2 u 2. Therefore we get the well known equation. 2ax = v2 −u2 2 a x = v 2 − u 2.
For instance. Scalar field. It can interpolate only for spin 0 states, therefore, equation of motion is ( + m2)ϕ = 0. Vector field. It can interpolate for spin 0 and spin 1 states. The corresponding projectors (you can easily convince yourself) are ΠLμν = ∂μ∂ν , and ΠTμν = ημν − ΠLμν = ημν − ∂μ∂ν .
In any school Physics course, the Newtonian equations of motion are very much a ‘stock’ item. Students learn the equations and are given a variety of problems that provide practice in determining which equation (s) to use to solve any particular problem. What is perhaps a little surprising is that in general, no one applies the well ...
Ai = Ai(qi, ˙ qi, t) and Bi = Bi(qi, ˙ qi, t) So now we have 2n equations that each equal a constant or initial condition. But by definition an integral of motion is an equation that only depends on the initial conditions - it doesn't change with time. Since all of these 2n equations are equal to an initial condition, then you have 2n ...