With the help of d'alembert's principle,how can lagrangr equation of motion be obtained
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The principle states that the sum of the differences between the forces acting on a system of mass particles and the time derivatives of the momenta of the system itself projected onto any virtual displacementconsistent with the constraints of the system is zero. Thus, in symbols d'Alembert's principle is written as following,
{\displaystyle \sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0,}
where :
{\displaystyle i}is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,{\displaystyle \mathbf {F} _{i}}is the total applied force (excluding constraint forces) on the {\displaystyle i}-th particle,{\displaystyle m_{i}\scriptstyle }is the mass of the {\displaystyle i}-th particle,{\displaystyle \mathbf {a} _{i}}is the acceleration of the {\displaystyle i}-th particle,{\displaystyle m_{i}\mathbf {a} _{i}} together as product represents the time derivative of the momentum of the {\displaystyle i}-th particle if the masses do not change with time, and{\displaystyle \delta \mathbf {r} _{i}}is the virtual displacement of the {\displaystyle i}-th particle, consistent with the constraints.
{\displaystyle \sum _{i}(\mathbf {F} _{i}-m_{i}\mathbf {a} _{i})\cdot \delta \mathbf {r} _{i}=0,}
where :
{\displaystyle i}is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,{\displaystyle \mathbf {F} _{i}}is the total applied force (excluding constraint forces) on the {\displaystyle i}-th particle,{\displaystyle m_{i}\scriptstyle }is the mass of the {\displaystyle i}-th particle,{\displaystyle \mathbf {a} _{i}}is the acceleration of the {\displaystyle i}-th particle,{\displaystyle m_{i}\mathbf {a} _{i}} together as product represents the time derivative of the momentum of the {\displaystyle i}-th particle if the masses do not change with time, and{\displaystyle \delta \mathbf {r} _{i}}is the virtual displacement of the {\displaystyle i}-th particle, consistent with the constraints.
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the main principle of that does the difference of between them and does not change
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