D'Alembert's Principle

This principle attributed to [wikipedia:Jean_le_Rond_d'Alembert D'Alembert] formed the basis for Lagrange to develop
his Analytical Mechanics. It is an interesting principle that is of wide practical applicability. For problems in statics it
can greatly simplify and replace tedious calculations of forces on loaded structures. For dynamics, it is replaced by the
Lagrangian approach to which it naturally leads.

Basically the principle eliminates the need to consider explicitly the forces that arise due to constraints. Thus
you can afford to be quite normal without any tension. Naturally puns intended.

Case of a single particle

Consider a single particle to begin with. It is say constrained to move on some surface in 3D space whose equation is

(1)
\begin{align} f(\mathbf{r},t)=0. \end{align}

The time dependence allows the surface to change with time. Note that we are not allowing the particle to change the
shape, the shape changes anyway. You may think of light particles, say chalk powder, on paper or handkerchief that you
are wiggling around, its your wiggling thats changing the shape. Newton's equation for this particle is

(2)
\begin{align} m \ddot {\mathbf{ r}} = \mathbf{ F+C} \end{align}

where $\mathbf{F}$ is the external force, which could be gravity for instance in the case of the chalk powder on
handkerchief, and $\mathbf{C}$ is the force that the surface exerts on the particle in order to keep it on itself, that is the force
of constraint. For instance, in the case of a simple pendulum, where a mass is suspended to a string, it is the tension in
the string that is constraining the particle to be at a constant distance from a fixed point. This is an unknown force. Thus
we have in total six unknown functions to be found, the three components of $\mathbf{C}$ and the three components
of $\mathbf{r}$, while we have only four equations, the constraint equation (1) and the three differential equations in (2).
This situation is due to there being many forces of constraints that are possible solutions. In particular, say we have found one such, then adding to it another force that is everywhere parallel to the surface will also do.

At this point we need to make an assumption to make $\mathbf{C}$ unique. We assume that $\mathbf{C}$ is perpendicular to the surface. This is equivalent and will eventually lead to D'Alembert's principle that constraint forces do no virtual work. Thus this amounts to
constraining the constraint forces to the form

(3)
\begin{align} \mathbf{C}=\lambda(t) \nabla f(\mathbf{r},t) \end{align}

where $\lambda(t)$ is an undetermined function, possibly of time, but not of position. The entire position dependence of $\mathbf{C}$
is due to fact that it is proportional to the normal of the surface. This is the usual "normal force" that we encounter in a first course on mechanics.
Now we have four unknowns and four equations and therefore the problem is solvable.

Now we show that the assumption made above implies that the forces of constraint do no work unless the constraints are time dependent.
Assume that the force $\mathbf{F}$ can be derived from a potential, i.e. $\mathbf{F}=-\nabla V(\mathbf{r},t)$, as is very often the
case. Then Newton's equation is

(4)
\begin{align} m \frac{d\mathbf{v}}{dt}= -\nabla V \cdot + \lambda(t) \nabla f \end{align}

Take the dot product of both sides with the velocity $\mathbf{v}$.
Note that the kinetic energy $T$ is

(5)
\begin{align} T= \frac{m}{2} \frac{d}{dt} v^2=\frac{m}{2} \frac{d}{dt} \mathbf{v\cdot v}= \mathbf{v}\cdot m \frac{d \mathbf{v}}{dt} \end{align}

and

(6)
\begin{align} \frac{dV}{dt} = \nabla V \cdot \mathbf{v} + \frac{\partial V }{\partial t} \end{align}
(7)
\begin{align} \frac{df}{dt} = \nabla f \cdot \mathbf{v} + \frac{\partial f }{\partial t}=0 \end{align}

as $f=0$.
Thus we get

(8)
\begin{align} \frac{dT}{dt}= -\frac{dV}{dt}+\frac{\partial V }{\partial t}-\lambda(t) \frac{\partial f }{\partial t} \end{align}

or written in terms of the total energy $E=T+V$

(9)
\begin{align} \frac{dE}{dt}= \frac{\partial V }{\partial t}-\lambda(t) \frac{\partial f }{\partial t} \end{align}

The first term is the work done by time dependent external forces, while the second is due to shifting
constraints. Thus if the constraints are not changing with time, they can do no work. Note that we are ruling
out forces of friction as these are tangential to the surface.

To get back to our original problem, now that we have 4 unknowns and 4 equations, we solve as follows:
At any time $t$ and position $\mathbf{r}$ there is a tangent vector to the surface of constraint.
In fact there are typically an infinite number of such tangent vectors, except when the dimension of the
surface is one. Say that $\mathbf{\tau}$ is one such tangent vector. Then

(10)
\begin{align} (m \ddot {\mathbf{ r}}- \mathbf{ F})\mathbf{\cdot \tau}=0, \end{align}

as

(11)
\begin{align} \mathbf{\tau \cdot C}= \mathbf{\tau \cdot} \lambda \nabla f =0 \end{align}

Note that the vector $\tau$ is calculated at a fixed time and position. We may interpret Eq. (10) as
the net work done by inertial and external forces and this is zero for such "displacements". Most textbooks
talk of "virtual displacements" which are infinitesimals in the direction of $\tau$. There is no need to do so,
if we recognize the constraint surface and can calculate the instantaneous tangents on it.

Eq.(11) gives us two equations and along with the constraint we can solve for the orbit $\mathbf{r}(t)$.
If we so wish we can find the constraint force from:

(12)
\begin{align} \mathbf{C}=m \ddot {\mathbf{ r}}- \mathbf{ F} \end{align}

Generalization to many particles

Let us take $N$ particles with $K$ constraints. We write the equations of motion and the constraints as

(13)
\begin{align} m \ddot {\mathbf{ r}_i} = \mathbf{ F}_i+\mathbf{C}_i \end{align}
(14)
\begin{array} {l} f_1(\mathbf{r_1,r_2,\ldots,r_N})=0\\ \vdots\\ f_K(\mathbf{r_1,r_2,\ldots,r_N})=0 \end{array}

Now there are several constraining surfaces and the generalization of Eq. (3) above is not obvious.
The condition of Eq. (3) is called "smoothness". In there are many body but only one constraint, say
many particles on a plane, then the evident smoothness condition is

(15)
\begin{align} \mathbf{C}_i= \lambda(t) \nabla_i f \end{align}

where the subscript indicates that the gradient is to be calculated with respect to the i-th particle coordinate for
its corresponding constraint force. The number of unknown function is then $3N+1$ and the number of equations
are the same. If the function $\lambda(t)$ changed from particle to particle then it leads to $4N$ unknown functions,
therefore this is not the case.

If each constraint results in such a force, then we get in the case
of many constraints:

(16)
\begin{align} \mathbf{C}_i= \sum_{k=1}^K \lambda_k(t) \nabla_i f_k \end{align}

How many equations and how many unknowns?
Show that this implies the generalization that the forces of constraints do no work if they are time independent.
Let $\tau_i$ be $N$ arbitrary vectors that satisfy the condition that

(17)
\begin{align} \sum_{i=1}^N \tau_i \cdot \nabla_i f_k =0, \;\;\; k=1,\ldots,K. \end{align}

Of the $3N$ arbitrary components of $\tau$ vectors these $K$ conditions imply that only $3N-K$ are
really independent. Using Eqs. (13),(16) and (17) then gives

(18)
\begin{align} \sum_{i=1}^N (m_i \ddot {\mathbf{ r}_i}- \mathbf{ F}_i)\mathbf{\cdot \tau_i}=0, \end{align}

which is called D'Alembert's principle.
This gives $3N-K$ equations and along with the $K$ constraint equations can be used to solve for the $3N$ unknown
components of the $\mathbf{r}_i$.

The geometric picture that we had for the one particle case has now been complicated, but as we will see later
is completely retained. The configuration "manifold" though is now often not visualizable.

Problems

  1. Derive the equations of motion for the planar simple pendulum using D'Alembet's principle.
  2. Do the same for "Atwood's" machine (two masses, m and M connected by a thread hung over a pulley).
  3. Ditto for a particle constrained to move inside the surface of a vertical cone whose half-angle is $\alpha$.
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