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 and mathematician

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 ''Encyclopé ...

, and Italian-French mathematician

Joseph Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaprinciple of virtual work from static to

dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...

s by introducing ''forces of inertia'' which, when added to the applied forces in a system, result in ''dynamic equilibrium''. D'Alembert's principle can be applied in cases of kinematic constraints that depend on velocities. 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. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

, and more general specification of the irreversibility is required.

Statement of the principle

The principle states that the sum of the differences between the

force In physics, a force is an influence that can cause an Physical object, object to change its velocity unless counterbalanced by other forces. In mechanics, force makes ideas like 'pushing' or 'pulling' mathematically precise. Because the Magnitu ...

s acting on a system of massive particles and the time

derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...

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,

\sum_i ( \mathbf F_i - m_i \dot\mathbf_i - \dot_i\mathbf_i)\cdot \delta \mathbf r_i = 0,

where: *

i

is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system, *

\mathbf _i

is the total applied force (excluding constraint forces) on the

i

-th particle, *

m_i

is the mass of the

i

-th particle, *

\mathbf v_i

is the velocity of the

i

-th particle, *

\delta \mathbf r_i

is the virtual displacement of the

i

-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 Lagrangiageneralized forces In analytical mechanics (particularly Lagrangian mechanics), generalized forces are conjugate to generalized coordinates. They are obtained from the applied forces , acting on a system that has its configuration defined in terms of generalized ...

\mathbf Q_j

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 ...

Derivations

General case with variable mass

The general statement of d'Alembert's principle mentions "the time

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

i

-th mass is the product of its mass and velocity:

\mathbf p_i = m_i \mathbf v_i

and its time derivative is

\dot_i = \dot_i \mathbf_i + m_i \dot_i.

In many applications, the masses are constant and this equation reduces to

\dot_i = m_i \dot_i = m_i \mathbf_i.

However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms

\dot_i \mathbf_i

and

m_i \dot_i

have to remain present, giving

\sum_ ( \mathbf _ - m_i \mathbf_i - \dot_i \mathbf_i)\cdot \delta \mathbf r_i = 0.

If the variable mass is ejected with a velocity

\mathbf_i

the principle has an additional term:

\sum_ ( \mathbf _ - m_i \mathbf_i - \dot_i (\mathbf_i - \mathbf_i))\cdot \delta \mathbf r_i = 0.

Special case with constant mass

Consider Newton's law for a system of particles of constant mass,

i

. The total force on each particle is

\mathbf _^ = m_i \mathbf _i,

where *

\mathbf _^

are the total forces acting on the system's particles, *

m_i \mathbf _i

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:

\mathbf _^ - m_i \mathbf _i = \mathbf 0.

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 fo ...

\delta W

, done by the total and inertial forces together through an arbitrary virtual displacement,

\delta \mathbf r_i

, of the system leads to a zero identity, since the forces involved sum to zero for each particle.

\delta W = \sum_ \mathbf _^ \cdot \delta \mathbf r_i - \sum_ m_i \mathbf_i \cdot \delta \mathbf r_i = 0

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,

\mathbf F_i

, and constraint forces,

\mathbf C_i

, yields

\delta W = \sum_ \mathbf _ \cdot \delta \mathbf r_i + \sum_ \mathbf _ \cdot \delta \mathbf r_i - \sum_ m_i \mathbf_i \cdot \delta \mathbf r_i = 0.

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,

\sum_ \mathbf _ \cdot \delta \mathbf r_i = 0

. 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:

\delta W = \sum_ ( \mathbf _ - m_i \mathbf_i )\cdot \delta \mathbf r_i = 0.

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, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial frame of reference. Unlike real forces, which result from physical ...

" and "inertial torque" 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''. Some educators caution that attempts to use d'Alembert inertial mechanics lead students to make frequent sign errors.Ruina, Andy L.
and

Rudra Pratap Rudra Pratap is an Indian academician and the vice-chancellor of Plaksha University, Mohali. Previously, he was a professor of Mechanical Engineering at the Indian Institute of Science (IISc), Bangalore. Among other research interests, he work ...

Introduction to statics and dynamics
Pre-print for Oxford University Press, 2008. A potential cause for these errors is the sign of the inertial forces. Inertial forces can be used to describe an apparent force in a

non-inertial reference frame A non-inertial reference frame (also known as an accelerated reference frame) is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-z ...

that has an acceleration

\mathbf

with respect to an

inertial reference frame In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...

. In such a non-inertial reference frame, a mass that is at rest and has zero acceleration in an inertial reference system, because no forces are acting on it, will still have an acceleration

-\mathbf

and an apparent inertial, or pseudo or

fictitious force A fictitious force, also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial reference frame, non-inertial frame of reference. Unlike real forc ...

-m\mathbf

will seem to act on it: in this situation the inertial force has a minus sign.

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

n

rigid bodies with

m

generalized coordinates requires

\delta W = \left(Q_1 + Q_1^*\right) \delta q_1 + \dots + \left(Q_m + Q_m^*\right) \delta q_m = 0,

for any set of virtual displacements

\delta q_j

with

Q_j

being a generalized applied force and

Q^*_j

being a generalized inertia force. This condition yields

m

equations,

Q_j + Q^*_j = 0, \quad j=1, \ldots, m,

which can also be written as

\frac \frac -\frac = Q_j, \quad j=1,\ldots,m.

The result is a set of m equations of motion that define the dynamics of the rigid body system.

Formulation using the Lagrangian

D'Alembert's principle can be rewritten in terms of the Lagrangian

L=T-V

of the system as a generalized version of

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 funct ...

for the case of point particles, as follows,

\delta \int_^ L(\mathbf, \dot,t) dt + \sum_i\int_^ \mathbf_i \cdot \delta \mathbf r_i dt= 0,

where: *

\mathbf=(\mathbf_1,..., \mathbf _N)

\mathbf_i

are the applied forces *

\delta \mathbf_i

is the virtual displacement of the

i

-th particle, consistent with the constraints

\sum_i\mathbf_i \cdot \delta \mathbf _i=0

* the critical curve satisfies the constraints

\sum_i\mathbf_i \cdot \dot \mathbf _i=0

With the Lagrangian

L(\mathbf, \dot,t) = \sum_i \frac m_i \dot _i^2,

the previous statement of d'Alembert principle is recovered.

Generalization for thermodynamics

An extension of d'Alembert's principle can be used in thermodynamics. For instance, for an adiabatically closed

thermodynamic system A thermodynamic system is a body of matter and/or radiation separate from its surroundings that can be studied using the laws of thermodynamics. Thermodynamic systems can be passive and active according to internal processes. According to inter ...

described by a Lagrangian depending on a single entropy ''S'' and with constant masses

m_i

, such as

L(\mathbf, \dot,S,t) = \sum_i \frac m_i \dot _i^2 - V(\mathbf,S),

it is written as follows

\delta \int_^ L(\mathbf, \dot,S,t) dt + \sum_i\int_^ \mathbf_i \cdot \delta \mathbf r_i dt= 0,

where the previous constraints

\sum_i\mathbf_i \cdot \delta \mathbf _i=0

and

\sum_i\mathbf_i \cdot \dot \mathbf _i=0

are generalized to involve the entropy as: *

\sum_i\mathbf_i \cdot \delta \mathbf _i+T \delta  S=0

\sum_i\mathbf_i \cdot \dot \mathbf _i+T \dot  S=0.

Here

T=\partial V/\partial S

is the temperature of the system,

\mathbf_i

are the external forces,

\mathbf_i

are the internal dissipative forces. It results in the mechanical and thermal balance equations:

m_i\mathbf_i=- \frac+ \mathbf_i+\mathbf_i, \;\;i=1,...,N \qquad \qquad T \dot  S = -\sum_i\mathbf_i \cdot \dot \mathbf _i.

Typical applications of the principle include thermo-mechanical systems, membrane transport, and chemical reactions. For

\delta S=\dot S=0

the classical d'Alembert principle and equations are recovered.

References

{{DEFAULTSORT:D'alembert'S Principle Classical mechanics Dynamical systems Lagrangian mechanics Principles