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These include differential equations, manifolds, Lie groups, and ergodic theory. [4] This article gives a summary of the most important of these. This article lists equations from Newtonian mechanics, see analytical mechanics for the more general formulation of classical mechanics (which includes Lagrangian and Hamiltonian mechanics).
Dynamical systems theory and chaos theory deal with the long-term qualitative behavior of dynamical systems.Here, the focus is not on finding precise solutions to the equations defining the dynamical system (which is often hopeless), but rather to answer questions like "Will the system settle down to a steady state in the long term, and if so, what are the possible steady states?", or "Does ...
The general equation can then be written as [6] = + + (),. where the "force" term corresponds to the forces exerted on the particles by an external influence (not by the particles themselves), the "diff" term represents the diffusion of particles, and "coll" is the collision term – accounting for the forces acting between particles in collisions.
The direct simulation Monte Carlo algorithm is like molecular dynamics in that the state of the system is given by the positions and velocities of the particles, {,}, for =, …,. Unlike molecular dynamics, each particle in a DSMC simulation represents F N {\displaystyle F_{N}} molecules in the physical system that have roughly the same ...
The difference between states A and B may be in the atom types involved, in which case the ΔF obtained is for "mutating" one molecule onto another, or it may be a difference of geometry, in which case one obtains a free-energy map along one or more reaction coordinates. This free-energy map is also known as a potential of mean force (PMF).
Free body and kinetic diagrams of an inclined block. In dynamics a kinetic diagram is a pictorial device used in analyzing mechanics problems when there is determined to be a net force and/or moment acting on a body. They are related to and often used with free body diagrams, but depict only the net force and moment rather than all of the ...
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.
In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces.The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference ...