Deriving the Common Form of the Lagrangian from D'Alembert's Principle.

Variational path for Lagrangian mechanics

D'Alembert's Principle may be take as the fundamental assumption of all of classical mechanics. Taking this as an axiom below, we derive the form of the Lagrangian.

This short and informal introduction gives a brief derivation of the form of the Lagrangian \(L=T-V\), building essentially from D'Alembert's principal to Hamilton's principle. It is intended to give a somewhat concise statement of the simplest form of these principles.

Here we only consider systems consisting of particles (as opposed to fields) and classical forces (as opposed to quantum or relativistic). Generally we will also assume the Lagrangian's dependence on time is only contributed through the path \(q(t)\) and it's velocity \(\dot{q}(t)\), and will denote this with the standard notation \(\frac{\partial L}{\partial t}=0\). We will further assume that the path taken, and it's first derivative with respect to time, are continuous.

Principle of Virtual Work

In a typical course covering static loads, many engineers are taught the principle of virtual work (a typical physicist may be less familiar with it). It will serve as our foundation for the discussion that follows.

The principle of virtual work states that in static equilibrium, all virtual displacements generating work according to the instantaneous configuration of forces sum to zero. That is, virtual work must be zero in configurations of static equilibrium.

\[\sum_i{F_i\cdot \delta \vec{\textbf{r}}_i}=0\]

Newtonian Relation

D'Alembert's principle also still requires the adoption of the Newtonian relation that forces correspond to a change in momentum , i.e. \(I=\frac{dp}{dt}\). And we will also assume applied forces to be the differential of some corresponding scalar function, \(F=-\frac{dV}{dr}\).

D'Alembert's Principle

The principle of virtual work for static equilibrium (\(\sum_i{F_i\cdot \delta \vec{\textbf{r}}_i}=0\)) can be adapted to dynamical systems by treating the reactions of massive bodies as 'inertial forces' exactly balancing imparted forces. This adapted principle can be described as below,

\[ \textit{D'Alembert's Principle:} \quad \sum_{i}{(F_{i}-I_{i})\cdot\delta \vec{\textbf{r}}_{i}}=0 \]
where \(F_i=F^{Applied}_i+f^{constraint}_{i}\) is the sum of applied and constraint forces on object \(i\); \(I_i=\frac{d}{dt}(m\vec{v})\) is the modeled inertial force for object \(i\); and \(\delta \vec{\textbf{r}}\) is the virtual displacement (virtual stipulating it occurs not over time but in some instant, so as to prevent conditions from changing). Assuming displacements happen "harmoniously" with respect to the constraints, i.e. the constraints are satisfied perfectly, the term in \(F_i\) accounting for constraints can generally be omitted from considerations, as it disappears independently with \(\sum_i{f^{constraint}_i}\cdot\delta\vec{\textbf{r}}=0\). Further, since conservative forces can be modeled as the differential of a scalar field, \(-\frac{dV}{d\vec{r}}=F\), the first term can be considered a direct variation in a suitable potential energy, \(\delta V\).
\[ \sum_{i}{(F_{i})\cdot\delta \vec{\textbf{r}}_{i}}=\sum_{i}{(-\frac{dV}{d\vec{r}})\cdot\delta \vec{\textbf{r}}_{i}}=\sum_{i}{-\delta V_i} \]
where the summation is still over \(i\) particles. We neglect to generalize coordinates and forces here as it is accomplished with a simple application of the chain rule.

When considering D'Alembert's principle, it's important to understand the concept of a virtual displacement. A virtual displacement is a change in the system that happens independent of time (as opposed to displacements caused by motion). So, virtual displacements at some instant test the configuration of forces at that instant.

Hamilton's Principle

A logically equivalent principle to D'Alembert's principle, and one which is extensible to dynamical systems over a given time or path, is Hamilton's principle. It may be considered as the minimization of an integral over some virtual displacement in path \(\delta\vec{\textbf{r}}\), where the variation (and it's derivative) is continuous and vanishes at the points denoted by \(t_a\) and \(t_b\).

\[ \textit{Hamilton's Principle:} \quad\int_{t_a}^{t_b}dt \sum_{i}{(F_{i}-I_{i})\cdot\delta \vec{\textbf{r}}_{i}}=0 \]
The second term here, \(I_i\cdot\delta \vec{\textbf{r}}\), can be expanded as \(\frac{d}{dt}(m\vec{v})\cdot\delta \vec{\textbf{r}}\) and by the product rule, we know
\[ \frac{d}{dt}(m\vec{v}\cdot \delta \vec{\textbf{r}})=\frac{d}{dt}(m\vec{v})\cdot \delta\vec{\textbf{r}}+m\vec{v}\cdot\delta\vec{\textbf{v}}. \] \[ \int_{t_a}^{t_b}dt\left[\frac{d}{dt}(m\vec{v})\cdot\delta \vec{\textbf{r}}\right] =\int_{t_a}^{t_b}dt\left[ \frac{d}{dt}(m\vec{v}\cdot \delta \vec{\textbf{r}})-m\vec{v}\cdot\delta\vec{\textbf{v}}\right] =[m\vec{v}\cdot \delta \vec{\textbf{r}}]\vert_{t_a}^{t_b}-\int_{t_a}^{t_b}dt[m\vec{v}\cdot\delta\vec{\textbf{v}}] \]
Recalling the property of \(\delta\vec{\textbf{r}}\) that requires it vanish at the boundaries \(t_a\) and \(t_b\), it follows that any dot product with it at those points vanishes as well so the first term here is zero. We can also develop \(\int_{t_a}^{t_b}dt[m\vec{v}\cdot\delta\vec{\textbf{v}}]\) into a variation of a scalar function of \(\vec{v}\) (namely a function of \(v^2\)) as follows.
\[ -\int_{t_a}^{t_b}[m\vec{v}\cdot\delta\vec{\textbf{v}}]dt=-\int_{t_a}^{t_b}\delta\left[ \frac{m}{2}\vec{v}\cdot\vec{v}\right] dt =-\delta\int_{t_a}^{t_b}Tdt \]
The scalar function \(T=\frac{1}{2}mv^2\) is defined as the kinetic energy of the particle. Here we have neglected the summation term due to the linearity of the integral. Thus, Hamilton's principle can be simplified as such,
\[ \delta\int_{t_a}^{t_b}(T-V)dt=0 \]
and we may further define the quantity \(T-V\) as the Lagrangian \(L=T-V\). This is the usual form seen in introductory mechanics courses.

Further Resources

Jessica Coopersmith's lovely The Lazy Universe gives the most philosophically motivated (if not just lengthy) development of the principle of stationary action from the principle of virtual work (derives form of L). However, it lacks the more modern mathematical machinery of variational mechanics.

Cornelius Lanczos' The Variational Principles of Mechanics is a classic introduction and relatively in-depth development of variational mechanics from the principle of virtual work (derives form of L). Good for guided mathematical development aswell as concurrent and independent philosophical considerations.

Landau and Lifshitz' first volume of several hefty physics textbooks Course in Theoretical Physics takes the principle of stationary action as a given but develops the beginnings of variational mechanics from there.

David Cline's Variational Principles in Classical Mechanics gives good exposition on the topic.

Goldstein, Poole, and Safko's Analytical Mechanics gives a brief development of D'Alembert's Principle and then independently considers Hamilton's Principle and the basic tools of variational mechanics. However, some may not be able to bear looking at the book for too long.

Davidson Soper's Classical Field Theory gives a development of variational mechanics in field theory from the particle paradigm.