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Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. They are named in honour of Leonhard Euler.
Assuming that the masses are constant, G is one-half the time derivative of this moment of inertia: = = = = = = =. In turn, the time derivative of G is = = + = = = + = = + =, where m k is the mass of the k th particle, F k = dp k / dt is the net force on that particle, and T is the total kinetic energy of the system according to the v k ...
This equation states that the kinetic energy (E k) is equal to the integral of the dot product of the momentum (p) of a body and the infinitesimal change of the velocity (v) of the body. It is assumed that the body starts with no kinetic energy when it is at rest (motionless).
The second part expresses the kinetic energy of a system of particles in terms of the velocities of the individual particles and the centre of mass.. Specifically, it states that the kinetic energy of a system of particles is the sum of the kinetic energy associated to the movement of the center of mass and the kinetic energy associated to the movement of the particles relative to the center ...
Turbulence kinetic energy is then transferred down the turbulence energy cascade, and is dissipated by viscous forces at the Kolmogorov scale. This process of production, transport and dissipation can be expressed as: D k D t + ∇ ⋅ T ′ = P − ε , {\displaystyle {\frac {Dk}{Dt}}+\nabla \cdot T'=P-\varepsilon ,} where: [ 1 ]
The formula for the acceleration A P can now be obtained as: = ˙ + + (), or = / + / +, where α is the angular acceleration vector obtained from the derivative of the angular velocity vector; / =, is the relative position vector (the position of P relative to the origin O of the moving frame M); and = ¨ is the acceleration of the origin of ...
This happens frequently in Hamiltonian mechanics, with T being the kinetic energy and V the potential energy. For the notational simplicity, let us introduce the symbol = (,) to denote the canonical coordinates including both the position and momentum coordinates. Then, the set of the Hamilton's equations given in the introduction can be ...