Lagrangian mechanics 729567 221183078 2008-06-23T12:02:18Z Wolfkeeper 41690 Revert to revision 220795268 dated 2008-06-21 16:46:09 by سعی using [[:en:Wikipedia:Tools/Navigation_popups|popups]] {{Classical mechanics|cTopic=Formulations}} '''Lagrangian mechanics''' is a re-formulation of [[classical mechanics]] that combines [[conservation of momentum]] with [[conservation of energy]]. It was introduced by [[Joseph Louis Lagrange]] in [[1788]]. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving Lagrange's equation, given herein, for each of the system's [[generalized coordinates]]. The [[fundamental lemma of calculus of variations]] shows that solving Lagrange's equation is equivalent to finding the path that minimizes the [[Action (physics)|action functional]], a quantity that is the [[integral]] of the [[Lagrangian]] over time. The use of generalized coordinates may considerably simplify a system's [[analysis]]. For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using [[Newtonian mechanics]] would require solving for the time-varying constraint force required to keep the bead in the groove. For the same problem using Lagrangian mechanics, one looks at the path of the groove and chooses a set of ''independent'' generalized coordinates that completely characterize the possible motion of the bead. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the groove on the bead at a given moment. == Lagrange's equations == The equations of motion in Lagrangian mechanics are ''Lagrange's equations'', also known as ''[[Euler–Lagrange equation]]s''. Below, we sketch out the derivation of Lagrange's equation. Please note that in this context, V is used rather than U for potential energy and T replaces K for kinetic energy. See the references for more detailed and more general derivations. Start with [[D'Alembert's principle]] for the [[virtual work]] of applied forces, <math>\mathbf{F}_i</math>, and [[inertia]]l forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints:<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 |language=English |isbn=0-03-063366-4 |chapter=Energy Methods}}</ref>{{rp|269}} :<math>\delta W = \sum_{i=1}^n ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0</math>. ::<math>\delta W</math> is the virtual work ::<math>\delta \mathbf r_i</math> is the virtual displacement of the system, consistent with the constraints ::<math>m_i</math> are the masses of the particles in the system ::<math>\mathbf a_i</math> are the accelerations of the particles in the system ::<math>m_i \mathbf a_i</math> together as products represent the time derivatives of the system momenta, aka. inertial forces ::<math>i</math> is an integer used to indicate (via subscript) a variable corresponding to a particular particle ::<math>n</math> is the number of particles under consideration Break out the two terms: :<math>\delta W = \sum_{i=1}^n \mathbf {F}_{i} \cdot \delta \mathbf r_i - \sum_{i=1}^n m_i \mathbf{a}_i \cdot \delta \mathbf r_i = 0</math>. Assume that the following transformation equations from m independent [[generalized coordinates]], <math>q_j</math>, hold:<ref name="Torby1984"/>{{rp|260}} :<math>\mathbf{r}_1=\mathbf{r}_1(q_1, q_2, ..., q_m, t)</math>, :<math>\mathbf{r}_2=\mathbf{r}_2(q_1, q_2, ..., q_m, t)</math>, ... :<math>\mathbf{r}_n=\mathbf{r}_n(q_1, q_2, ..., q_m, t)</math>. ::<math>m</math> (without a subscript) indicates the total number generalized coordinates An expression for the [[virtual displacement]] (differential), <math>\delta \mathbf{r}_i</math>, of the system is<ref name="Torby1984"/>{{rp|264}} :<math>\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j</math>. ::<math>j</math> is an integer used to indicate (via subscript) a variable corresponding to a generalized coordinate The applied forces may be expressed in the generalized coordinates as [[generalized forces]], <math>Q_j</math>,<ref name="Torby1984"/>{{rp|265}} :<math>Q_j = \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j}</math>. Combining the equations for <math>\delta W</math>, <math>\delta \mathbf{r}_i</math>, and <math>Q_j</math> yields the following result after pulling the sum out of the dot product in the second term:<ref name="Torby1984"/>{{rp|269}} :<math>\delta W = \sum_{j=1}^m Q_j \delta q_j - \sum_{j=1}^m \sum_{i=1}^n m_i \mathbf{a}_i \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j = 0</math>. Substituting in the result from the [[#Kinetic energy relations|kinetic energy relations]] to change the inertial forces into a function of the kinetic energy leaves<ref name="Torby1984"/>{{rp|270}} :<math>\delta W = \sum_{j=1}^m Q_j \delta q_j - \sum_{j=1}^m \left ( \frac {d}{d t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) - \frac {\partial T}{\partial q_j} \right ) \delta q_j = 0</math>. In the above equation, <math>\delta q_j</math> is arbitrary, though it is—by definition—consistent with the constraints. So the relation must hold term-wise:<ref name="Torby1984"/>{{rp|270}} :<math>Q_j = \frac {d}{d t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) - \frac {\partial T}{\partial q_j}</math>. If the <math>\mathbf F_i</math> are conservative, they may be represented by a [[scalar potential]] field, <math>V</math>:<ref name="Torby1984"/>{{rp|266 & 270}} <math>\mathbf F_i = - \nabla V \Rightarrow Q_j = - \sum_{i=1}^n \nabla V \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} = - \frac {\partial V}{\partial q_j}</math>. The previous result may be easier to see by recognizing that <math>V</math> is a function of the <math>\mathbf {r}_i</math>, which are in turn functions of <math>q_j</math>, and then applying the [[chain rule]] to the derivative of <math>V</math> with respect to <math>q_j</math>. The definition of the [[Lagrangian]] is<ref name="Torby1984"/>{{rp|270}} :<math>\mathcal{L} = T - V</math>. Since the potential field is only a function of position, not velocity, '''Lagrange's equations''' are as follows:<ref name="Torby1984"/>{{rp|270}} :<math>0 = \frac {d}{d t} \left ( \frac {\partial \mathcal L}{\partial \dot{q}_j} \right ) - \frac {\partial \mathcal L}{\partial q_j}</math>. This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to <math>\dot{q}_j</math> and time, and solely with respect to <math>q_j</math>, adding the results and associating terms with the equations for <math>\mathbf F_i </math> and <math> Q_j </math>. In a more general formulation, the forces could be both potential and [[viscosity|viscous]]. If an appropriate transformation can be found from the <math>\mathbf F_i</math>, [[John Strutt, 3rd Baron Rayleigh|Rayleigh]] suggests using a dissipation function, <math>D</math>, of the following form:<ref name="Torby1984"/>{{rp|271}} :<math>D = \frac {1}{2} \sum_{j=1}^m \sum_{k=1}^m C_{j k} \dot{q}_j \dot{q}_k</math>. ::<math>C_{j k}</math> are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them If <math>D</math> is defined this way, then<ref name="Torby1984"/>{{rp|271}} :<math>Q_j = - \frac {\partial V}{\partial q_j} - \frac {\partial D}{\partial \dot{q}_j}</math> and :<math>0 = \frac {d}{d t} \left ( \frac {\partial \mathcal L}{\partial \dot{q}_j} \right ) - \frac {\partial \mathcal L}{\partial q_j} + \frac {\partial D}{\partial \dot{q}_j}</math>. ===Kinetic energy relations=== The [[kinetic energy]], <math>T</math>, for the system of particles is defined by<ref name="Torby1984"/>{{rp|269}} :<math>T = \frac {1}{2} \sum_{i=1}^n m_i \mathbf {v}_i \cdot \mathbf {v}_i</math>. The partial derivative of <math>T</math> with respect to the [[time derivative]]s of the generalized coordinates, <math>\dot{q}_j</math>, is<ref name="Torby1984"/>{{rp|269}} :<math>\frac {\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf {v}_i \cdot \frac {\partial \mathbf {v}_i} {\partial \dot{q}_j}</math>. The previous result may be difficult to visualize. As a result of the [[product rule]], the derivative of a general [[dot product]] <math>\frac{d}{dx} ( \mathbf{f}(x) \cdot \mathbf{g}(x) )</math> is <math>\mathbf{f}(x) \cdot \frac{d}{dx} \mathbf{g}(x) + \mathbf{g}(x) \cdot \frac{d}{dx} \mathbf{f}(x)</math> This general result may be seen by briefly stepping into a [[Cartesian coordinate system]], recognizing that the dot product is (there) a term-by-term product sum, and also recognizing that the derivative of a sum is the sum of its derivatives. In our case, '''f''' and '''g''' are equal to '''v''', which is why the factor of one half disappears. According to the [[chain rule]] and the coordinate transformation equations given above for <math>\mathbf{r}</math>, it's time derivative, <math>\mathbf{v}</math>, is:<ref name="Torby1984"/>{{rp|264}} :<math>\mathbf{v}_i = \sum_{j=1}^m \frac {\partial \mathbf{r}_i}{\partial q_j} \dot{q}_j + \frac {\partial \mathbf{r}_i}{\partial t}</math>. Together, the definition of <math>\mathbf v_i</math> and the total differential, <math>d \mathbf {r}_i</math>, suggest that<ref name="Torby1984"/>{{rp|269}} :<math>\frac {\partial \mathbf {v}_i}{\partial \dot{q}_j} = \frac {\partial \mathbf {r}_i}{\partial q_j}</math>.{{Clarifyme|date=March 2008}} [ Remember that :<math> \frac {\partial } {\partial {\dot{q}_k}} {A}{\dot{q}_k} = A </math>, and it is easier to visualise the result if you replace the subscript <math>j</math> with some other subscript <math>k</math>. Also remember that in the sum, there is only one <math> {\dot{q}_k} </math>. ] Substituting this relation back into the expression for the partial derivative of <math>T</math> gives<ref name="Torby1984"/>{{rp|269}} :<math>\frac {\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf v_i \cdot \frac {\partial \mathbf {r}_i}{\partial q_j}</math>. Taking the time derivative gives<ref name="Torby1984"/>{{rp|270}} :<math>\frac {d}{d t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) = \sum_{i=1}^n \left [ m_i \mathbf a_i \cdot \frac {\partial \mathbf {r}_i}{\partial q_j} + m_i \mathbf {v}_i \cdot \frac {d}{d t} \left ( \frac {\partial \mathbf {r}_i}{\partial q_j} \right ) \right ]</math>. Using the chain rule on the last term gives<ref name="Torby1984"/>{{rp|270}} :<math>\frac {d}{d t} \left ( \frac {\partial \mathbf {r}_i}{\partial q_j} \right ) = \sum_{k=1}^m \frac {\partial^2 \mathbf r_i}{\partial q_j \partial q_k} \dot{q_k} + \frac {\partial^2 \mathbf r_i}{\partial q_j \partial t}</math>. From the expression for <math>\mathbf v_i</math>, one sees that<ref name="Torby1984"/>{{rp|270}} :<math>\frac {d}{d t} \left ( \frac {\partial \mathbf {r}_i}{\partial q_j} \right ) = \frac {\partial \mathbf {v}_i}{\partial q_j}</math>. This allows simplification of the last term,<ref name="Torby1984"/>{{rp|270}} :<math>\frac {d}{d t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) = \sum_{i=1}^n \left [ m_i \mathbf a_i \cdot \frac {\partial \mathbf {r}_i}{\partial q_j} + m_i \mathbf {v}_i \cdot \frac {\partial \mathbf {v}_i}{\partial q_j} \right ]</math>. The partial derivative of <math>T</math> with respect to the generalized coordinates, <math>q_j</math>, is<ref name="Torby1984"/>{{rp|270}} :<math>\frac {\partial T}{\partial q_j} = \sum_{i=1}^n m_i \mathbf {v}_i \cdot \frac {\partial \mathbf {v}_i} {\partial q_j}</math>.{{Clarifyme|date=March 2008}} [This last result may be obtained by doing a partial differentiation directly on the kinetic energy definition represented by the first equation.] The last two equations may be combined to give an expression for the inertial forces in terms of the kinetic energy:<ref name="Torby1984"/>{{rp|270}} :<math>\frac {d}{d t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right ) - \frac {\partial T}{\partial q_j} = \sum_{i=1}^n m_i \mathbf a_i \cdot \frac {\partial \mathbf {r}_i}{\partial q_j}</math> ==Old Lagrange's equations== Consider a single particle with [[mass]] ''m'' and [[position vector]] <math>\bold{r}</math>, moving under an applied [[force]], <math>\bold{F}</math>, which can be expressed as the [[gradient]] of a scalar potential energy function <math>V (\bold{r},t)</math>: :<math>\bold{F} = - \bold{\nabla} V.</math> Such a force is independent of third- or higher-order derivatives of <math>\bold{r}</math>, so [[Newton's laws of motion|Newton's second law]] forms a set of 3 second-order [[ordinary differential equation]]s. Therefore, the motion of the particle can be completely described by 6 independent variables, or ''degrees of freedom''. An obvious set of variables is <math>\{ \bold{r}_j, \dot{\bold{r}}_j | j = 1, 2, 3\}</math>, the Cartesian components of <math>\bold{r}</math> and their time derivatives, at a given instant of time (i.e. position (x,y,z) and velocity <math>(v_x,v_y,v_z)</math>). More generally, we can work with a set of [[generalized coordinates]], <math>q_j</math>, and their time derivatives, the [[Generalized coordinates#Generalized velocities and kinetic energy|generalized velocities]], <math>\dot{q_j}</math>. The position vector, <math>\bold{r}</math>, is related to the generalized coordinates by some ''transformation equation'': :<math>\bold{r} = \bold{r}(q_i , q_j , q_k, t).</math> For example, for a [[simple pendulum]] of length ''l'', a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be :<math>\bold{r}(\theta, \dot{\theta} , t) = (l \sin \theta, l \cos \theta)</math>. The term "generalized coordinates" is really a holdover from the period when [[Cartesian coordinate system|Cartesian coordinates]] were the default coordinate system. Consider an arbitrary displacement <math>\delta \bold{r}</math> of the particle. The [[Mechanical work|work]] done by the applied force <math>\bold{F}</math> is <math>W = \bold{F} \cdot \delta \bold{r}</math>. Using Newton's second law, we write: :<math>\begin{matrix} \bold{F} \cdot \delta \bold{r} = m\ddot{\bold{r}} \cdot \delta \bold{r}. \end{matrix}</math> Since work is a physical scalar quantity, we should be able to rewrite this equation in terms of the generalized coordinates and velocities. On the left hand side, :<math> \begin{matrix} \bold{F} \cdot \bold{\delta} \bold{r} & = & - \bold{\nabla} V \cdot \displaystyle\sum_i {\partial \bold{r} \over \partial q_i} \delta q_i \\ \\ & = & - \displaystyle\sum_{i,j} {\partial V \over \partial r_j} {\partial r_j \over \partial q_i} \delta q_i \\ \\ & = & - \displaystyle\sum_i {\partial V \over \partial q_i} \delta q_i. \\ \end{matrix} </math> On the right hand side, carrying out a change of coordinates{{Clarifyme|date=March 2008}}, we obtain: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_{i,j} \ddot{r_i} {\partial r_i \over \partial q_j} \delta q_j </math> Rearranging Slightly: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \ddot{r_i} {\partial r_i \over \partial q_j} \right] \delta q_j </math> Now, by performing an "integration by parts" transformation, with respect to t: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} \left( \dot{r_i} {\partial r_i \over \partial q_j} \right) - \dot{r_i} {\mathrm{d} \over \mathrm{d}t}\left( {\partial r_i \over \partial q_j} \right) \right] \right] \delta q_j </math> Recognizing that <math>{\mathrm{d} \over \mathrm{d}t}{\partial r_j \over \partial q_i} = {\partial \dot{r_j} \over \partial q_i}</math> and <math>{\partial r_j \over \partial q_i} = {\partial \dot{r_j} \over \partial \dot{q_i}}</math>, we obtain: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} \left( \dot{r_i} {\partial \dot{r_i} \over \partial \dot{q_j}} \right) - \dot{r_i} {\partial \dot{r_i} \over \partial q_j} \right] \right] \delta q_j </math> Now, by changing the order of differentiation, we obtain: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} {\partial \over \partial \dot{q_j}} \left( \frac{1}{2} \dot{r_i}^2 \right) - {\partial \over \partial q_j} \left( \frac{1}{2} \dot{r_i}^2 \right) \right] \right] \delta q_j </math> Finally, we change the order of summation: <math>m \ddot{\bold{r}} \cdot \delta \bold{r} = \sum_j \left[ {\mathrm{d} \over \mathrm{d}t} {\partial \over \partial \dot{q_j}} \left( \sum_i \frac{1}{2} m \dot{r_i}^2 \right) - {\partial \over \partial q_j} \left( \sum_i \frac{1}{2} m \dot{r_i}^2 \right) \right] \delta q_j </math> Which is equivalent to: :<math> m \ddot{\bold{r}} \cdot \delta \bold{r} = \sum_i \left[{\mathrm{d} \over \mathrm{d}t}{\partial T \over \partial \dot{q_i}}-{\partial T \over \partial q_i}\right]\delta q_i </math> where <math>T=\frac{1}{2}m\dot{\bold{r}}\cdot\dot{\bold{r}}</math> is the kinetic energy of the particle. Our equation for the work done becomes :<math> \sum_i \left[{\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{\dot{q_i}}}-{\partial{(T-V)}\over \partial q_i}\right] \delta q_i = 0. </math> However, this must be true for ''any'' set of generalized displacements <math>\delta q_i</math>, so we must have :<math> \left[ {\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{\dot{q_i}}}-{\partial{(T-V)}\over \partial q_i}\right] = 0 </math> for ''each'' generalized coordinate <math>\delta q_i</math>. We can further simplify this by noting that ''V'' is a function solely of '''r''' and ''t'', and '''r''' is a function of the generalized coordinates and ''t''. Therefore, ''V'' is independent of the generalized velocities: :<math> {\mathrm{d} \over \mathrm{d}t}{\partial{V}\over \partial{\dot{q_i}}} = 0. </math> Inserting this into the preceding equation and substituting ''L'' = ''T'' - ''V'', called the Lagrangian, we obtain Lagrange's equations: :<math> {\partial{\mathcal{L}}\over \partial q_i} = {\mathrm{d} \over \mathrm{d}t}{\partial{\mathcal{L}}\over \partial{\dot{q_i}}}. </math> There is one Lagrange equation for each generalized coordinate q<sub>i</sub>. When q<sub>i</sub> = r<sub>i</sub> (i.e. the generalized coordinates are simply the Cartesian coordinates), it is straightforward to check that Lagrange's equations reduce to Newton's second law. The above derivation can be generalized to a system of ''N'' particles. There will be 6''N'' generalized coordinates, related to the position coordinates by 3''N'' transformation equations. In each of the 3''N'' Lagrange equations, ''T'' is the total kinetic energy of the system, and ''V'' the total potential energy. In practice, it is often easier to solve a problem using the [[Euler–Lagrange equation]]s than Newton's laws. This is because appropriate generalized coordinates ''q''<sub>i</sub> may be chosen to exploit symmetries in the system. ==Examples== In this section two examples are provided in which the above concepts are applied. The first example establishes that in a simple case, the Newtonian approach and the Lagrangian formalism agree. The second case illustrates the power of the above formalism, in a case which is hard to solve with Newton's laws. ===Falling mass=== Consider a point mass ''m'' falling freely from rest. By gravity a force ''F = m g'' is exerted on the mass (assuming ''g'' constant during the motion). Filling in the force in Newton's law, we find <math>\ddot x = g</math> from which the solution :<math>x(t) = \frac{1}{2} g t^2</math> follows (choosing the origin at the starting point). This result can also be derived through the Lagrange formalism. Take ''x'' to be the coordinate, which is ''0'' at the starting point. The kinetic energy is <math>T = \frac{1}{2} m v^2</math> and the potential energy is <math>V = - m g x</math>, hence :<math>\mathcal{L} = T - V = \frac{1}{2} m \dot{x}^2 + m g x</math>. Now we find :<math>0 = \frac{\partial \mathcal{L}}{\partial x} - \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial \dot x} = m g - m \frac{\mathrm{d} \dot x}{\mathrm{d} t} </math> which can be rewritten as <math>\ddot x = g</math>, yielding the same result as earlier. ===Pendulum on a movable support=== Consider a pendulum of mass ''m'' and length ''l'', which is attached to a support with mass ''M'' which can move along a line in the ''x''-direction. Let ''x'' be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle ''θ'' from the vertical. The kinetic energy can then be shown to be :<math>T = \frac{1}{2} M \dot{x}^2 + \frac{1}{2} m \left( \dot{x}_\mathrm{pend}^2 + \dot{y}_\mathrm{pend}^2 \right) = \frac{1}{2} M \dot{x}^2 + \frac{1}{2} m \left[ \left( \dot x + l \dot\theta \cos \theta \right)^2 + \left( l \dot\theta \sin \theta \right)^2 \right], </math> and the potential energy of the system is :<math> V = m g \operatorname{y}_\mathrm{pend} = - m g l \cos \theta . </math> [[Image:pendulumWithMovableSupport.svg|thumb|right|Sketch of the situation with definition of the coordinates (click to enlarge)]] Now carrying out the differentiations gives for the support coordinate ''x'' :<math>\frac{\mathrm{d}}{\mathrm{d}t} \left[ (M + m) \dot x + m l \dot\theta \cos\theta \right] = 0, </math> therefore: :<math> (M + m) \ddot x + m l \ddot\theta\cos\theta-m l \dot\theta ^2 \sin\theta = 0 </math> indicating the presence of a constant of motion. The other variable yields :<math>\frac{\mathrm{d}}{\mathrm{d}t}\left[ m( l^2 \dot\theta + \dot x l \cos\theta ) \right] + m (\dot x l \dot \theta + g l) \sin\theta = 0</math>; therefore :<math>\ddot\theta + \frac{\ddot x}{l} \cos\theta + \frac{g}{l} \sin\theta = 0 </math>. These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much harder and prone to errors. By considering limit cases (<math>\ddot x \to 0</math> should give the equations of motion for a pendulum, <math>\ddot\theta \to 0</math> should give the equations for a pendulum in a constantly accelerating system, etc.) the correctness of this system can be verified. == Hamilton's principle == The action, denoted by <math>\mathcal{S}</math>, is the time integral of the Lagrangian: :<math>\mathcal{S} = \int \mathcal{L}\,\mathrm{d}t.</math> Let ''q<sub>0</sub>'' and ''q<sub>1</sub>'' be the coordinates at respective initial and final times ''t<sub>0</sub>'' and ''t<sub>1</sub>''. Using the [[calculus of variations]], it can be shown the Lagrange's equations are equivalent to ''[[Hamilton's principle]]'': :''The system undergoes the trajectory between t<sub>0</sub> and t<sub>1</sub> whose action has a stationary value.'' By ''stationary'', we mean that the action does not vary to first-order for infinitesimal deformations of the trajectory, with the end-points (''q<sub>0</sub>'', ''t<sub>0</sub>'') and (''q<sub>1</sub>'',''t<sub>1</sub>'') fixed. Hamilton's principle can be written as: :<math>\delta \mathcal{S} = 0. \,\!</math> Thus, instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action. Hamilton's principle is sometimes referred to as the ''[[principle of least action]]''. However, this is a misnomer: the action only needs to be stationary, and the correct trajectory could be produced by a maximum, [[saddle point]], or minimum in the action. We can use this principle instead of [[Newton's Laws]] as the fundamental principle of mechanics, this allows us to use an integral principle (Newton's Laws are based on differential equations so they are a differential principle) as the basis for mechanics. However it is not widely stated that Hamilton's principle is a variational principle only with [[holonomic]] constraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving [[d'Alembert]] principle of [[virtual work]]. Working only with holonomic constraints is the price we have to pay for using an elegant variational formulation of mechanics. == Extensions of Lagrangian mechanics == The [[Hamiltonian mechanics|Hamiltonian]], denoted by ''H'', is obtained by performing a [[Legendre transformation]] on the Lagrangian. The Hamiltonian is the basis for an alternative formulation of classical mechanics known as [[Hamiltonian mechanics]]. It is a particularly ubiquitous quantity in [[quantum mechanics]] (see [[Hamiltonian (quantum mechanics)]]). In [[1948]], [[Richard Feynman|Feynman ]] invented the [[path integral formulation]] extending the [[principle of least action]] to [[quantum mechanics]] for [[electrons]] and [[photons]]. In this formulation, particles travel every possible path between the initial and final states; the probability of a specific final state is obtained by summing over all possible trajectories leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and [[Fermat's principle]] in [[optics]]. ==See also== *[[Restricted three-body problem]] *[[Hamiltonian mechanics]] *[[Functional derivative]] *[[Nielsen form]] *[[Canonical coordinates]] *[[Generalized coordinates]] *[[Lagrangian analysis]] (applications of Lagrangian mechanics) == References == <references/> * Goldstein, H. ''Classical Mechanics,'' second edition, pp.16 (Addison-Wesley, 1980) * Moon, F. C. ''Applied Dynamics With Applications to Multibody and Mechatronic Systems'', pp. 103-168 (Wiley, 1998). ==Further reading== * [[Lev Landau|Landau, L.D.]] and [[Evgeny Lifshitz|Lifshitz, E.M.]] ''Mechanics'', Pergamon Press. * Gupta, Kiran Chandra, ''Classical mechanics of particles and rigid bodies'' (Wiley, 1988). ==External links== * Tong, David, [http://www.damtp.cam.ac.uk/user/tong/dynamics.html Classical Dynamics] Cambridge lecture notes * [http://www.eftaylor.com/software/ActionApplets/LeastAction.html Principle of least action interactive] Excellent interactive explanation/webpage * [http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-61Aerospace-DynamicsSpring2003/D453E02B-5218-4154-8531-DB35ECD76A6C/0/lecture9.pdf Aerospace dynamics lecture notes on Lagrangian mechanics] * [http://ocw.mit.edu/NR/rdonlyres/Aeronautics-and-Astronautics/16-61Aerospace-DynamicsSpring2003/53F21B11-4F88-4870-967A-0C05AD85B104/0/lecture10.pdf Aerospace dynamics lecture notes on Rayleigh dissipation function] [[Category:Lagrangian mechanics|*]] [[ar:ميكانيك لاغرانج]] [[ca:Formulació lagrangiana]] [[cs:Lagrangeovská formulace mechaniky]] [[de:Lagrange-Formalismus]] [[es:Mecánica lagrangiana]] [[fa:مکانیک لاگرانژی]] [[fr:Équations de Lagrange]] [[ko:라그랑주 역학]] [[id:Mekanika Lagrangian]] [[it:Meccanica lagrangiana]] [[ja:ラグランジュ力学]] [[no:Lagrangemekanikk]] [[ru:Лагранжева механика]] [[sl:Lagrangeeva formulacija gibalnih enačb]] [[fi:Lagrangen mekaniikka]] [[sv:Euler-Lagranges ekvationer]] [[zh:拉格朗日力学]]