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Classical mechanics utilises many equations—as well as other mathematical concepts—which relate various physical quantities to one another. These include differential equations, manifolds, Lie groups, and ergodic theory. [4] This article gives a summary of the most important of these.
The radius of this circle, , can be determined by equating the magnitude of the Lorentz force to the centripetal force as = | |. Rearranging, the gyroradius can be expressed as = | |. Thus, the gyroradius is directly proportional to the particle mass and perpendicular velocity, while it is inversely proportional to the particle electric charge ...
In three spatial dimensions, this is a system of three coupled second-order ordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vector r of the particle at time t , subject to the initial conditions of r and v when t = 0.
For rod length 6" and crank radius 2" (as shown in the example graph below), numerically solving the acceleration zero-crossings finds the velocity maxima/minima to be at crank angles of ±73.17615°. Then, using the triangle law of sines, it is found that the rod-vertical angle is 18.60647° and the crank-rod angle is 88.21738°. Clearly, in ...
Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility.In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges.
The Wigner–Seitz radius, named after Eugene Wigner and Frederick Seitz, is the radius of a sphere whose volume is equal to the mean volume per atom in a solid (for first group metals). [1] In the more general case of metals having more valence electrons, r s {\\displaystyle r_{\\rm {s}}} is the radius of a sphere whose volume is equal to the ...
Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, so that some coordinate does not occur in the Hamiltonian (i.e. a cyclic coordinate), the corresponding momentum coordinate is conserved along each trajectory, and that coordinate can be reduced to a constant in the other equations of the set.
Stated another way, Lambert's problem is the boundary value problem for the differential equation ¨ = ^ of the two-body problem when the mass of one body is infinitesimal; this subset of the two-body problem is known as the Kepler orbit.