# Derivation of Lagrange Equations from Newton's Second Law

This question somehow stems from the concepts of analytical mechanics and this question of mine in Physics SE which did not receive a good answer. I will try to re-phrase the Physics SE post in pure mathematical terms so anyone interested in physical backgrounds may read that post. Please feel free to edit the question if my presentation seems to be vague.

In what follows, the indices $i$, $j$ are positive integers that vary as $i=1,\dots,R$, $j=1,\dots,M$, where $R$ and $M$ are given positive integers such that $R\le M$.

Suppose that $q_j:[0,\infty)\to\mathbb{R}$ are some $C^1[0,\infty)$ non constant functions named as generalized coordinates. Also, $\mathcal{F}_j:[0,\infty)\to\mathbb{R}$ are some other functions. Theses two sets of functions are related to each other by

$$\sum_{j=1}^{M}\mathcal{F}_j(t)\dot q_j(t)=0,\quad \forall t \in [0,\infty),\tag{1}$$

where $\dot q_j$ are derivatives of $q_j$. Suppose that following equations hold

\begin{align*} \sum_{j=1}^{M}a_{ij}(t)\dot q_j(t)+b_i(t\big)=0,\quad \forall t\in[0,\infty), \tag{2} \end{align*}

where $a_{ij}:[0,\infty)\to\mathbb{R}$ and $b_i:[0,\infty)\to\mathbb{R}$ are given functions. Also, the rank of the matrix $A(t)=[a_{ij}(t)]_{R\times M}$ is $R$ for each $t$. These equations are known as the linear generalized velocity constraints. Also, the set of functions $\big\{q_1,\dots,q_n\big\}$ with $n=M-R$ is linearly independent. $n$ resembles the degree of freedom of the system of particles.

The goal is to show that \begin{align*} &\mathcal{F}_j\big(t\big)=\sum_{i=1}^{R}\lambda_i(t)a_{ij}(t) \tag{3} \end{align*} where $\lambda_i:[0,\infty)\to\mathbb{R}$ are some functions.

My Thought

It seems that Eq.$(3)$ is implying that the vector $\boldsymbol{\mathcal{F}}(t)$ lies in the row space of the matrix $A(t)$.

The simplest case is to consider that there were no constraint equations then $R=0,\,n=M$ and Eq.$(4)$ reduces to $F_j(t)=0$. It is seen that Eq. $(1)$ implies this result. The key point is that since $\big\{q_1,\dots,q_n\big\}$ is linearly independent and $q_j$ are non-constant functions so $\big\{\dot q_1,\dots,\dot q_n\big\}$ will be linearly independent.

I have no clue to go further.

Any hint or help is appreciated.

• Can you explain a bit of what the result is supposed to mean? It is a bit opaque as it stands. – Ian Oct 19 '17 at 15:27
• @Ian: Thanks for the attention. :) Please take a look at the question linked in Physics SE and let me know if there are any ambiguity. Compare the first equation here with equation (7) in the Physics SE post and the second equation with equations (2) in Physics SE post. Please feel free to make any improvement as you think is required. – Hosein Rahnama Oct 19 '17 at 15:39
• You've chosen odd variable names, etc... The way I've always proven that equivalence is backwards - from the $\frac{d}{dt} \frac{\partial \mathcal L}{\partial \dot q} = \frac{\partial \mathcal L}{\partial q}$ back to $\mathbf F = m \mathbf a$. There's one step that doesn't work backwards that easily, but that's the procedure. – user121330 Oct 19 '17 at 16:03