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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.
MPM can be implemented to solve either quasi-static or dynamic equations of motion, depending on the type of problem that is to be modeled. Several versions of MPM include Generalized Interpolation Material Point Method [33];Convected Particle Domain Interpolation Method; [34] Convected Particle Least Squares Interpolation Method. [35]
where denotes the surface normal direction of the solid structure, and the imaginary particle density is calculated through the equation of state. Accordingly, the interaction forces f a F : p {\displaystyle \mathbf {f} _{a}^{F:p}} and f a F : v {\displaystyle \mathbf {f} _{a}^{F:v}} acting on a solid particle a {\displaystyle a} are given by
The equation is a nonlinear integro-differential equation, and the unknown function in the equation is a probability density function in six-dimensional space of a particle position and momentum. The problem of existence and uniqueness of solutions is still not fully resolved, but some recent results are quite promising. [3] [4]
The original Langevin equation [1] [2] describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid, = + (). Here, v {\displaystyle \mathbf {v} } is the velocity of the particle, λ {\displaystyle \lambda } is its damping coefficient, and m {\displaystyle m} is its mass.
The small particle size also implies that the disturbed flow can be found in the limit of very small Reynolds number, leading to a drag force given by Stokes' drag. Unsteadiness of the flow relative to the particle results in force contributions by added mass and the Basset force. The BBO equation states:
Standard Model of Particle Physics. The diagram shows the elementary particles of the Standard Model (the Higgs boson, the three generations of quarks and leptons, and the gauge bosons), including their names, masses, spins, charges, chiralities, and interactions with the strong, weak and electromagnetic forces.
This is an example of an equation that holds off shell, since it is true for any fields configuration regardless of whether it respects the equations of motion (in this case, the Euler–Lagrange equation given above). However, we can derive an on shell equation by simply substituting the Euler–Lagrange equation: