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There is a highly ambitious article by Udwadia and Kalaba: article . To explain their main point, I will first recall the D'Alembert-Lagrange principle.
Assume we have have a mechanical system of particles with masses ##m_1,\ldots, m_\nu## and with position vectors ##\boldsymbol r_1,\ldots, \boldsymbol r_\nu.## The ideal constraints are imposed
$$f_s(t,\boldsymbol r_1,\ldots, \boldsymbol r_\nu,\boldsymbol {\dot r}_1,\ldots, \boldsymbol {\dot r}_\nu)=0,\quad s=1,\ldots,n<m:=3\nu.\qquad (1)$$
The space of virtual displacements is defined as follows
$$\Delta=\Big\{(\delta \boldsymbol r_1,\ldots,\delta \boldsymbol r_\nu)\in\mathbb{R}^m\Big|
\sum_{i=1}^\nu\Big(\frac{\partial f_s}{\partial\boldsymbol{\dot r}_i},\delta \boldsymbol r_i\Big)=0,\quad s=1,\ldots,n\Big\}.$$
The standard and important condition is as follows:
$$\dim \Delta=m-n.\qquad (2)$$
This condition means that the system of constraints (1) is non-degenerate. Under this assumption we can prove that the D'Alembert-Lagrange principle
$$\sum_{i=1}^\nu(m_i\boldsymbol {\ddot r}_i-\boldsymbol F_i,\delta \boldsymbol r_i)=0,\quad (\delta \boldsymbol r_1,\ldots,\delta \boldsymbol r_\nu)\in\Delta$$ is correct: given the initial positions and velocities, the solution exists, is unique, and depends continuously on the initial conditions.
The authors of the article just discard condition (2) and claim that they have discovered "a new fundamental view of Lagrangian mechanics." Nice try!
Assume we have have a mechanical system of particles with masses ##m_1,\ldots, m_\nu## and with position vectors ##\boldsymbol r_1,\ldots, \boldsymbol r_\nu.## The ideal constraints are imposed
$$f_s(t,\boldsymbol r_1,\ldots, \boldsymbol r_\nu,\boldsymbol {\dot r}_1,\ldots, \boldsymbol {\dot r}_\nu)=0,\quad s=1,\ldots,n<m:=3\nu.\qquad (1)$$
The space of virtual displacements is defined as follows
$$\Delta=\Big\{(\delta \boldsymbol r_1,\ldots,\delta \boldsymbol r_\nu)\in\mathbb{R}^m\Big|
\sum_{i=1}^\nu\Big(\frac{\partial f_s}{\partial\boldsymbol{\dot r}_i},\delta \boldsymbol r_i\Big)=0,\quad s=1,\ldots,n\Big\}.$$
The standard and important condition is as follows:
$$\dim \Delta=m-n.\qquad (2)$$
This condition means that the system of constraints (1) is non-degenerate. Under this assumption we can prove that the D'Alembert-Lagrange principle
$$\sum_{i=1}^\nu(m_i\boldsymbol {\ddot r}_i-\boldsymbol F_i,\delta \boldsymbol r_i)=0,\quad (\delta \boldsymbol r_1,\ldots,\delta \boldsymbol r_\nu)\in\Delta$$ is correct: given the initial positions and velocities, the solution exists, is unique, and depends continuously on the initial conditions.
The authors of the article just discard condition (2) and claim that they have discovered "a new fundamental view of Lagrangian mechanics." Nice try!
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