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| {{Classical mechanics|cTopic=Fundamental concepts}}
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| [[Image:Alembert.jpg|thumb|right|[[Jean d'Alembert]]]]
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| '''D'Alembert's principle''', also known as the '''Lagrange–d'Alembert principle''', is a statement of the fundamental [[classical physics|classical]] laws of motion. It is named after its discoverer, the [[France|French]] [[physicist]] and [[mathematician]] [[Jean le Rond d'Alembert]]. It is the dynamic analogue to the ''principle of [[virtual work]] for applied forces'' in a static system and in fact is more general than [[Hamilton's principle]], avoiding restriction to [[holonomic system]]s.<ref name=Lanczos>{{cite book |last=Lanczos |first=Cornelius |title=The Variational Principles of Mechanics |edition=4th |year=1970 |isbn=0-486-65067-7 |publisher=Dover Publications Inc. |location=New York |url=http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA92&dq=%22d%27Alembert%27s+principle%22 |page=92}}</ref> A holonomic constraint depends only on the coordinates and time. It does not depend on the velocities. If the negative terms in accelerations are recognized as ''[[inertial force]]s'', the statement of d'Alembert's principle becomes ''The total virtual work of the impressed forces plus the inertial forces vanishes for reversible displacements''.<ref name=Lanczos2>{{cite book |title=p. 90 |author=Cornelius Lanczos |isbn=0-486-65067-7 |year=1970 |url=http://books.google.com/books?id=ZWoYYr8wk2IC&printsec=frontcover&dq=%22d%27Alembert%27s+principle%22#PPA90,M1}}</ref>
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| The principle states that the sum of the differences between the [[force]]s acting on a system of mass particles and the time [[derivative]]s of the [[momentum|momenta]] of the system itself along any [[virtual displacement]] consistent with the constraints of the system, is zero. Thus, in symbols d'Alembert's principle is written as following,
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| :<math>\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0,</math>
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| where
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| :{|
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| | <math>i</math> || is an integer used to indicate (via subscript) a variable corresponding to a particular particle in the system,
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| | <math>\mathbf {F}_i</math> || is the total applied force (excluding constraint forces) on the <math>i</math>-th particle,
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| | <math> m_i \scriptstyle</math> || is the mass of the <math>i</math>-th particle,
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| | <math>\mathbf a_i</math> || is the acceleration of the <math>i</math>-th particle,
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| | <math>m_i \mathbf a_i</math> || together as product represents the time derivative of the momentum of the <math>i</math>-th particle, and
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| | <math>\delta \mathbf r_i</math> || is the virtual displacement of the <math>i</math>-th particle, consistent with the constraints.
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| |}
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| This above equation is often called d'Alembert's principle, but it was first written in this variational form by [[Joseph Louis Lagrange]].<ref>Arnold Sommerfeld (1956), ''Mechanics: Lectures on Theoretical Physics'', Vol 1, p. 53</ref> D'Alembert's contribution was to demonstrate that in the totality of a dynamic system the forces of constraint vanish. That is to say that the [[generalized forces]] <math>{\mathbf Q}_{j}</math> need not include constraint forces.
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| ==General case with changing masses==
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| The general statement of d'Alembert's principle mentions "the time [[derivative]]s of the [[momentum|momenta]] of the system". The momentum of the ''i''-th mass is the product of its mass and velocity:
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| :<math>\mathbf p_i = m_i \mathbf v_i</math>
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| and its time derivative is
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| :<math>\dot{\mathbf{p}_i} = \dot{m}_i \mathbf{v}_i + m_i \dot{\mathbf{v}}_i</math>.
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| In many applications, the masses are constant and this equation reduces to
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| :<math>\dot{\mathbf{p}_i} = m_i \dot{\mathbf{v}}_i = m_i \mathbf{a}_i</math>,
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| which appears in the formula given above. However, some applications involve changing masses (for example, chains being rolled up or being unrolled) and in those cases both terms <math>\dot{m}_i \mathbf{v}_i</math> and <math>m_i \dot{\mathbf{v}}_i</math> have to remain present, giving
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| :<math>\sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i - \dot{m}_i \mathbf{v}_i)\cdot \delta \mathbf r_i = 0.</math>
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| ==Derivation for special cases==
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| To date nobody has shown that D'Alembert's principle is equivalent to Newton's Second Law. This is true only for some very special cases e.g. rigid body constraints. However, an approximate solution to this problem does exist.<ref name="Rebhan2006">{{cite book |last=Rebhan |first=Eckhard |title=Mechanik |series=Theoretische Physik |year=2006 |publisher=Spektrum Akademischer Verlag |location=Heidelberg, Germany |isbn=978-3-8274-1716-9 |chapter=Exkurs 5.1: Ableitung des d'Alembert Prinzips}}</ref>
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| Consider Newton's law for a system of particles, i. The total force on each particle is<ref name="Torby1984">{{cite book |last=Torby |first=Bruce |title=Advanced Dynamics for Engineers |series=HRW Series in Mechanical Engineering |year=1984 |publisher=CBS College Publishing |location=United States of America |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>
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| :<math>\mathbf {F}_{i}^{(T)} = m_i \mathbf {a}_i,</math>
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| where
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| :{|
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| | <math>\mathbf {F}_{i}^{(T)}</math> || are the total forces acting on the system's particles,
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| | <math>m_i \mathbf {a}_i</math> || are the inertial forces that result from the total forces.
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| |}
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| 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:<ref name="Torby1984"/>
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| :<math>\mathbf {F}_{i}^{(T)} - m_i \mathbf {a}_i = \mathbf 0.</math>
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| Considering the [[virtual work]], <math>\delta W</math>, done by the total and inertial forces together through an arbitrary virtual displacement, <math>\delta \mathbf r_i</math>, of the system leads to a zero identity, since the forces involved sum to zero for each particle.<ref name="Torby1984"/>
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| :<math>\delta W = \sum_{i} \mathbf {F}_{i}^{(T)} \cdot \delta \mathbf r_i - \sum_{i} m_i \mathbf{a}_i \cdot \delta \mathbf r_i = 0</math>
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| 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, <math>\mathbf F_i</math>, and constraint forces, <math>\mathbf C_i</math>, yields<ref name="Torby1984"/>
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| :<math>\delta W = \sum_{i} \mathbf {F}_{i} \cdot \delta \mathbf r_i + \sum_{i} \mathbf {C}_{i} \cdot \delta \mathbf r_i - \sum_{i} m_i \mathbf{a}_i \cdot \delta \mathbf r_i = 0.</math>
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| 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 do no work. Such displacements are said to be ''consistent'' with the constraints.<ref name="Jong2005">{{cite web |url=http://people.eng.unimelb.edu.au/montyjp/Mechanics/05Portland.pdf |title=Teaching Students Work and Virtual Work Method in Statics:A Guiding Strategy with Illustrative Examples |accessdate=2007-09-24 |author=Ing-Chang Jong |year=2005 |format=PDF |work=Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition |publisher=American Society for Engineering Education }}</ref> 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:.<ref name="Torby1984"/>
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| :<math>\delta W = \sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0.</math>
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| There is also a corresponding principle for static systems called the [[virtual work#Principle of virtual work for applied forces|principle of virtual work for applied forces]].
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| ==D'Alembert's principle of inertial forces==
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| D'Alembert showed that one can transform an accelerating rigid body into an equivalent static system by adding the so-called "[[inertial force]]" 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''.
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| ===Example for plane 2D motion of a rigid body===
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| For a planar rigid body, moving in the plane of the body (the ''x''–''y'' plane), and subjected to forces and torques causing rotation only in this plane, the inertial force is
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| :<math> \mathbf{F}_i = - m\ddot{\mathbf{r}}_c</math> | |
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| where <math>\mathbf{r}_c</math> is the position vector of the centre of mass of the body, and <math>m</math> is the mass of the body. The inertial torque (or moment) is
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| :<math>T_i = -I\ddot{\theta}</math> | |
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| where <math>I</math> is the [[moment of inertia]] of the body. If, in addition to the external forces and torques acting on the body, the inertia force acting through the center of mass is added and the inertial torque is added (acting around the centre of mass is as good as anywhere) the system is equivalent to one in static equilibrium. Thus the equations of static equilibrium
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| :<math>
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| \begin{align}
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| \sum F_x &= 0,
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| \\
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| \sum F_y &= 0,
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| \\
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| \sum T &= 0
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| \end{align}
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| </math>
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| hold. The important thing is that <math>\sum T</math> is the sum of torques (or moments, including the inertial moment and the moment of the inertial force) taken about ''any'' point. The direct application of Newton's laws requires that the angular acceleration equation be applied ''only'' about the center of mass.
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| ===Dynamic equilibrium===
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| 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 that
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| :<math> \delta W = (Q_1 + Q^*_1)\delta q_1 + \ldots + (Q_m + Q^*_m)\delta q_m = 0,</math> | |
| for any set of virtual displacements δq<sub>j</sub>. This condition yields m equations,
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| :<math> Q_j + Q^*_j = 0, \quad j=1, \ldots, m,</math>
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| which can also be written as
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| :<math> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = Q_j, \quad j=1,\ldots,m.</math>
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| The result is a set of m equations of motion that define the dynamics of the rigid body system.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:D'alembert'S Principle}}
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| [[Category:Classical mechanics]]
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| [[Category:Dynamical systems]]
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| [[Category:Lagrangian mechanics]]
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| [[Category:Principles]]
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