The D'Alembert-Lagrange Principle

The D'Alembert-Lagrange principle is one of the basic principles of mechanics , according to which, if inertia forces are attached to given (active) forces acting on points of a mechanical system, then when a mechanical system with ideal bonds moves at any moment of time, the sum of elementary workings of active forces and elementary work of inertia forces on any possible (virtual) movement of the system is zero ^[1]

The D'Alembert-Lagrange principle is a combination of the principle of possible displacements of statics and the D'Alembert principle of dynamics. Its use allows one to study the motion of mechanical systems with perfect bonds without introducing unknown coupling reactions into the equations of motion.

Conclusion

Let a mechanical system with holonomic, retaining, ideal bonds be represented by material points with masses $m_{1},m_{2},...,m_{N}$ ${\ displaystyle m_ {1}, m_ {2}, ..., m_ {N}}$ ^[2] . Let to each material point $m_{i}$ ${\ displaystyle m_ {i}}$ active forces are applied with the resultant ${\vec {F_{i}}}$ ${\ displaystyle {\ vec {F_ {i}}}}$ and passive forces with the resultant ${\vec {N_{i}}}$ ${\ displaystyle {\ vec {N_ {i}}}}$ . According to Newton’s second law :

m_{i}{\vec {w_{i}}}={\vec {F_{i}}}+{\vec {N_{i}}}

{\ displaystyle m_ {i} {\ vec {w_ {i}}} = {\ vec {F_ {i}}} + {\ vec {N_ {i}}}}

or

{\vec {F_{i}}}-m_{i}{\vec {w_{i}}}+{\vec {N_{i}}}=0

{\ displaystyle {\ vec {F_ {i}}} - m_ {i} {\ vec {w_ {i}}} + {\ vec {N_ {i}}} = 0}

(one)

We now fix a certain moment in time and let the mechanical system know the virtual (possible) movement $\delta {\vec {r_{1}}},\delta {\vec {r_{2}}},...,\delta {\vec {r_{N}}}$ ${\ displaystyle \ delta {\ vec {r_ {1}}}, \ delta {\ vec {r_ {2}}}, ..., \ delta {\ vec {r_ {N}}}}$ . Scalarly multiply each equation (1) by the corresponding $\delta {\vec {r_{i}}}$ ${\ displaystyle \ delta {\ vec {r_ {i}}}}$ and summarize all the equations:

\sum _{i=1}^{N}({\vec {F_{i}}}-m_{i}{\vec {w_{i}}})\delta {\vec {r_{i}}}+\sum _{i=1}^{N}{\vec {N_{i}}}\delta {\vec {r_{i}}}=0

{\ displaystyle \ sum _ {i = 1} ^ {N} ({\ vec {F_ {i}}} - m_ {i} {\ vec {w_ {i}}} \ delta {\ vec {r_ { i}}} + \ sum _ {i = 1} ^ {N} {\ vec {N_ {i}}} \ delta {\ vec {r_ {i}}} = 0}

The sum of perfect connections at any virtual displacement is zero, therefore:

\sum _{i=1}^{N}({\vec {F_{i}}}-m_{i}{\vec {w_{i}}})\delta {\vec {r_{i}}}=0

{\ displaystyle \ sum _ {i = 1} ^ {N} ({\ vec {F_ {i}}} - m_ {i} {\ vec {w_ {i}}} \ delta {\ vec {r_ { i}}} = 0}

This equality is called the general equation of mechanics .

In any mechanical system with ideal holding connections at any moment of time of movement on any virtual movement, the sum of the mechanical work performed by active and inertial forces is always zero.

Notes

↑ Targ S. M. D'Alembert - Lagrange principle // Physics. Encyclopedia / ed. A. M. Prokhorova - M., Big Russian Encyclopedia, 2003 .-- ISBN 5-85270-306-0 . - with. 142
↑ Bugaenko G.A., Malanin V.V. , Yakovlev V.I. Fundamentals of classical mechanics. - M., Higher School, 1999. - ISBN 5-06-003587-5 . - with. 218

[Targ-1] Targ S. M. D'Alembert - Lagrange principle // Physics. Encyclopedia / ed. A. M. Prokhorova - M., Big Russian Encyclopedia, 2003 .-- ISBN 5-85270-306-0 . - with. 142

[2] Bugaenko G.A., Malanin V.V. , Yakovlev V.I. Fundamentals of classical mechanics. - M., Higher School, 1999. - ISBN 5-06-003587-5 . - with. 218

The D'Alembert-Lagrange Principle

Conclusion

See also

Notes

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