Why can the LHS and RHS of the Euler-Lagrange equation be interpreted as "MASS TIMES ACCELERATION" and "FORCE" respectively?

jfondrie

Since L is a measure of energy, if we represent the RHS as being in units of Nxm, then differentiating wrt q gives us N, or units of force. On the LHS, if we represent energy as kgxm^2/s^2, then when we first differentiate wrt q dot (which is itself m/s), then we get kgxm/s or mass times velocity. Doing the final differentiation wrt dt then makes it kgxm/s^2, or mass times acceleration.

billiam

Since L is a scalar quantity, how can taking the partial derivative yield a vector quantity? I think there is something more fundamental here that I am not understanding.

tracychen

same confusion here, can anyone explain in more details?

motoole2

@billiam It doesn't; taking the partial derivative of a scalar function L produces a scalar function. However, we have a system of Euler-Lagrangian equations to consider, where we have N equations for an N-dimensional vector q (i.e., one equation for each component).

A-star

Hi. Can someone point me to more resources on this topic? It still looks confusing to me. Thanks.

Why can the LHS and RHS of the Euler-Lagrange equation be interpreted as "MASS TIMES ACCELERATION" and "FORCE" respectively?

Since L is a measure of energy, if we represent the RHS as being in units of Nxm, then differentiating wrt q gives us N, or units of force. On the LHS, if we represent energy as kgxm^2/s^2, then when we first differentiate wrt q dot (which is itself m/s), then we get kgxm/s or mass times velocity. Doing the final differentiation wrt dt then makes it kgxm/s^2, or mass times acceleration.

Since L is a scalar quantity, how can taking the partial derivative yield a vector quantity? I think there is something more fundamental here that I am not understanding.

same confusion here, can anyone explain in more details?

@billiam It doesn't; taking the partial derivative of a scalar function L produces a scalar function. However, we have a

systemof Euler-Lagrangian equations to consider, where we have N equations for an N-dimensional vector`q`

(i.e., one equation for each component).Hi. Can someone point me to more resources on this topic? It still looks confusing to me. Thanks.