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D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental
classical laws of motion. It is named after its discoverer, the
French physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe.
Physicists generally are interested in the root or ultimate cau ...
and
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert (; ; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was, together with Denis Diderot, a co-editor of the '' Encyclopéd ...
. D'Alembert's principle generalizes the
principle of virtual work
A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule that has to be or usually is to be followed. It can be desirably followed, or it can be an inevitable consequence of something, such as the la ...
from
static
Static may refer to:
Places
*Static Nunatak, a nunatak in Antarctica
United States
* Static, Kentucky and Tennessee
*Static Peak, a mountain in Wyoming
**Static Peak Divide, a mountain pass near the peak
Science and technology Physics
*Static el ...
to
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in ...
s by introducing ''forces of inertia'' which, when added to the applied forces in a system, result in ''dynamic equilibrium''.
The principle does not apply for irreversible displacements, such as sliding
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
, and more general specification of the irreversibility is required. D'Alembert's principle is more general than
Hamilton's principle
In physics, Hamilton's principle is William Rowan Hamilton's formulation of the principle of stationary action. It states that the dynamics of a physical system are determined by a variational problem for a functional based on a single function, ...
as it is not restricted to
holonomic constraint
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form:
:f(u_1, u_2, u_3,\ldots, u_n, t) = 0
where \ are the ''n'' generalized coordinates that d ...
s that depend only on coordinates and time but not on velocities.
Statement of the principle
The principle states that the sum of the differences between the
forces acting on a system of massive particles and the time
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of the
momenta of the system itself projected onto any
virtual displacement
In analytical mechanics, a branch of applied mathematics and physics, a virtual displacement (or infinitesimal variation) \delta \gamma shows how the mechanical system's trajectory can ''hypothetically'' (hence the term ''virtual'') deviate very ...
consistent with the constraints of the system is zero. Thus, in mathematical notation, d'Alembert's principle is written as follows,
where:
*
is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,
*
is the total applied force (excluding constraint forces) on the
-th particle,
*
is the mass of the
-th particle,
*
is the velocity of the
-th particle,
*
is the virtual displacement of the
-th particle, consistent with the constraints.
Newton's dot notation is used to represent the derivative with respect to time. The above equation is often called d'Alembert's principle, but it was first written in this variational form by
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[generalized forces Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, Fi, i=1,..., n, acting on a system that has its configuration defined in terms of generaliz ...](_blank)
need not include constraint forces. It is equivalent to the somewhat more cumbersome
Gauss's principle of least constraint
The principle of least constraint is one variational formulation of classical mechanics enunciated by Carl Friedrich Gauss in 1829, equivalent to all other formulations of analytical mechanics. Intuitively, it says that the acceleration of a c ...
.
Derivations
General case with variable mass
The general statement of D'Alembert's principle mentions "the time
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of the
momenta of the system." By Newton's second law, the first time derivative of momentum is the force. The momentum of the
-th mass is the product of its mass and velocity:
and its time derivative is
In many applications, the masses are constant and this equation reduces to
However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms
and
have to remain present, giving
Special case with constant mass
Consider Newton's law for a system of particles of constant mass,
. The total force on each particle is
where
*
are the total forces acting on the system's particles,
*
are the inertial forces that result from the total forces.
Moving the inertial forces to the left gives an expression that can be considered to represent quasi-static equilibrium, but which is really just a small algebraic manipulation of Newton's law:
Considering the
virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for ...
,
, done by the total and inertial forces together through an arbitrary virtual displacement,
, of the system leads to a zero identity, since the forces involved sum to zero for each particle.
The original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the total forces into applied forces,
, and constraint forces,
, yields
If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces (which is not usually the case, so this derivation works only for special cases), the constraint forces don't do any work,
. Such displacements are said to be ''consistent'' with the constraints.
This leads to the formulation of ''d'Alembert's principle'', which states that the difference of applied forces and inertial forces for a dynamic system does no virtual work:
There is also a corresponding principle for static systems called the
principle of virtual work for applied forces.
D'Alembert's principle of inertial forces
D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "
inertial force
A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.
It is related to Newton's second law of motion, which trea ...
" and "
inertial torque
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration ...
" or moment. The inertial force must act through the center of mass and the inertial torque can act anywhere. The system can then be analyzed exactly as a static system subjected to this "inertial force and moment" and the external forces. The advantage is that in the equivalent static system one can take moments about any point (not just the center of mass). This often leads to simpler calculations because any force (in turn) can be eliminated from the moment equations by choosing the appropriate point about which to apply the moment equation (sum of moments = zero). Even in the course of Fundamentals of Dynamics and Kinematics of machines, this principle helps in analyzing the forces that act on a link of a mechanism when it is in motion. In textbooks of engineering dynamics, this is sometimes referred to as ''d'Alembert's principle''.
Dynamic equilibrium
D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of
rigid bodies with
generalized coordinates requires
for any set of virtual displacements
with
being a
generalized applied force and
being a generalized inertia force. This condition yields
equations,
which can also be written as
The result is a set of m equations of motion that define the dynamics of the rigid body system.
References
{{DEFAULTSORT:D'alembert'S Principle
Classical mechanics
Dynamical systems
Lagrangian mechanics
Principles