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can include transverse waves that solve the equation central to Sturm–Liouville theory: − d d x [ τ ( x ) d ρ ( x ) d x ] + v ( x ) ρ ( x ) = ω 2 σ (...

and eigenfunctions of differential equations. For example, the Sturm–Liouville problem arises in the study of the harmonics of waves in a violin string...

Important extensions and analyses were contributed subsequently by Lagrange, Liouville, Laplace, Jacobi, Darboux,[citation needed] Le Verrier, Velde,[citation...

N-particle system in absence of quantum fluctuations is given by the Liouville equation for the probability density function f N = f N ( q 1 … q N ,...

two-body system under the influence of torque turns out to be a Sturm-Liouville equation. Although the two-body model treats the objects as point particles...

effect of two causes: the net balance of all external forces acting onto that object — magnitude is directly proportional to this net resulting force; that...

accessible microstates are equiprobable over a long period of time. Liouville's theorem states that, for a Hamiltonian system, the local density of microstates...

}\right)\cos(\omega t)=0.} This boundary condition is in the form of the Sturm–Liouville formulation. The latter boundary condition is satisfied when sin ⁡ ( 2...

the newton-metre (N⋅m). For more on the units of torque, see § Units. The net torque on a body determines the rate of change of the body's angular momentum...

motion of a simple pendulum, although for it to be an accurate model, the net force on the object at the end of the pendulum must be proportional to the...

on a dense subset of L2: P* is a densely defined operator. The Sturm–Liouville operator is a well-known example of a formal self-adjoint operator. This...

consecutive random variables (here the random variable is defined using the Liouville function (which is a useful mathematical function for studying distribution...

except insofar as it is acted upon by a force. At any instant of time, the net force on a body is equal to the body's acceleration multiplied by its mass...

Another similar equation comes from the Euler–Lagrange equation for Liouville field theory φ x x − φ t t = 2 e 2 φ . {\displaystyle \varphi _{xx}-\varphi...

motion states that the rate of change of a body's momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any...

undergoing motion. It quantifies both the distance and direction of the net or total motion along a straight line from the initial position to the final...

pseudovector. For rigid bodies, angular acceleration must be caused by a net external torque. However, this is not so for non-rigid bodies: For example...

not experience any net force will continue to move in a straight line with a constant velocity until they are subjected to a net force. Under everyday...

J. Ståhlberg, the first President of Finland (b. 1832) 1882 – Joseph Liouville, French mathematician and academic (b. 1809) 1894 – Hermann von Helmholtz...

There is a net force on the stone in the horizontal plane which acts toward the center. In an inertial frame of reference, were it not for this net force acting...