Newton's Equations state, $\overrightarrow { F } =\overrightarrow { \dot { p } } $ , and we can treat variable mass system using Newton's equations.

  1. While deriving the Lagrange Equations from d'Alembert's principle, I have seen that we treat the mass of the $i^{th}$ to be constant. Is this a necessary condition for Lagrangian formalism to hold? Is there a way to derive Lagrange equations without assuming this using d'Alembert's principle?

  2. Also, where is this information considered when we derive the Euler-Lagrange equations from an action principle?


Yes. You need to look at the general form of the Euler-Lagrange equations.

The Euler-Lagrange equations are simply, $$\begin{align} \frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot{x}_i} = \frac{\partial \mathcal{L}}{\partial x_i} \end{align}$$

The term I you care about is $ p_i = \frac{\partial \mathcal{L}}{\partial \dot{x}_i} $ which I call $ p $ because it is the canonical momentum. If you use the standard kinetic energy, this term becomes,

$$\begin{align} p_i = \frac{\partial \mathcal{L}}{\partial \dot{x}_i} = m_i \dot{x}_i \end{align}$$

Take the time derivative of that and in general, we get, $$\begin{align} \dot{p}_i = m_i \ddot{x}_i + \dot{m}_i \dot{x}_i \end{align}$$

  • $\begingroup$ When we derive the equations from d'Alembert's principle, where do we see this explicitly? $\endgroup$ – Abhikumbale Jan 9 '18 at 17:34
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    $\begingroup$ This doesn't answer the question in the last paragraph of the OP $\endgroup$ – DanielC Jan 9 '18 at 17:34
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    $\begingroup$ @DanielC Where variable mass is considered in deriving the Euler-Lagrange equations? As far as I know, it never was. The EL equations don't even require that the thing be a particle, let alone have a mass. $\endgroup$ – aidan.plenert.macdonald Jan 9 '18 at 18:07
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    $\begingroup$ @Abhikumbale Not entirely sure what you mean, but perhaps this helps. The Euler-Lagrange equations come from stationarity of the action. D'Alembert's principle is a bit different and permits non-holonomic systems. $\endgroup$ – aidan.plenert.macdonald Jan 9 '18 at 18:08
  • $\begingroup$ @aidan.plenert.macdonald thanks for the source, but can we derive a similar equation(like EL equations) for the changing mass ? $\endgroup$ – Abhikumbale Jan 9 '18 at 18:25

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