# Does Lagrangian Formalism Apply To Systems With Variable Mass?

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}

• When we derive the equations from d'Alembert's principle, where do we see this explicitly? – Abhikumbale Jan 9 '18 at 17:34
• This doesn't answer the question in the last paragraph of the OP – DanielC Jan 9 '18 at 17:34
• @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. – aidan.plenert.macdonald Jan 9 '18 at 18:07
• @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. – aidan.plenert.macdonald Jan 9 '18 at 18:08
• @aidan.plenert.macdonald thanks for the source, but can we derive a similar equation(like EL equations) for the changing mass ? – Abhikumbale Jan 9 '18 at 18:25