D'Alembert's principle

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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 caus ...
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 ...
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 ''Encyclopédi ...
. D'Alembert's principle generalizes the principle of virtual work from static 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 as it is not restricted to holonomic constraints 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
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s 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 Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles. In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ... of the system itself projected onto any virtual displacement 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 $\mathbf Q_j$ need not include constraint forces. It is equivalent to the somewhat more cumbersome Gauss's principle of least constraint. # 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 Momenta is an autonomous driving company headquartered in Beijing, China that aims to build the 'Brains' for autonomous vehicles. In December 2021, Momenta and BYD established a 100 million yuan ($15.7 million) joint venture to deploy autonomous ...
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 $\sum_ ( \mathbf _ - m_i \mathbf_i - \dot_i \mathbf_i)\cdot \delta \mathbf r_i = 0.$

## 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, $\delta W$, 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 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 tre ...
" 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''.

# 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.

# References

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