Euler andra lag
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Euler's laws of motion
Extend Newton's laws of motion to rigid bodies
"Euler's first law" and "Euler's second law" redirect here. For other uses, see Euler (disambiguation).
In classical mechanics, Euler's laws of motion are equations of motion which extend Newton's laws of motion for point particle to rigid body motion.[1] They were formulated by Leonhard Euler about 50 years after Isaac Newton formulated his laws.
Overview
[edit]Euler's first law
[edit]Euler's first law states that the rate of change of linear momentump of a rigid body is equal to the resultant of all the external forces Fext acting on the body:[2]
Internal forces between the particles that make up a body do not contribute to changing the momentum of the body as there is an equal and opposite force resulting in no net effect.[3]
The linear momentum of a rigid body is the product of the mass of the body and the velocity of its center of massvcm.[1][4][5]
Euler's second law
[edit]Main article: Balance of angular momentum
Euler's second law states that the rate of change of angular momentumL about a point that
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Euler-Lagrange Differential Equation
The Euler-Lagrange differential equation is the fundamental equation of calculus of variations. It states that if is defined by an integralof the form
(1) |
where
(2) |
then has a stationary value if the Euler-Lagrange differential equation
(3) |
is satisfied.
If time-derivativenotation is replaced instead by space-derivative notation , the equation becomes
(4) |
The Euler-Lagrange differential equation is implemented as [f, u[x], x] in the Wolfram Language package .
In many physical problems, (the partial derivative of with respect to ) turns out to be 0, in which case a manipulation of the Euler-Lagrange differential equation reduces to the greatly simplified and partially integrated form known as the Beltrami identity,
(5) |
For three independent variables (Arfken , pp. ), the equation generalizes to
(6) |
Problems in the calculus of variations often can be solved by solution of the appropriate Euler-Lagrange equation.
To derive the Euler-Lagrange differential equation, examine
since . Now, integrate the second
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Euler–Lagrange equation
Second-order partial differential equation describing motion of mechanical system
In the calculus of variations and classical mechanics, the Euler–Lagrange equations[1] are a struktur of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation fryst vatten useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This fryst vatten analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative fryst vatten zero. In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system fryst vatten described bygd the solutions to the Euler equation for the action of the struktur. In this context Euler equations are usually called Lagrange equations. In classical mechanics,[2] it is equivalent to Newton's laws