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Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11 Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial 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) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics 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 Fundamental concepts equations. There are fewer equations since one is not directly calculating the Space · Time · Velocity · Speed · Mass · influence of the groove on the bead at a given moment. Acceleration · Gravity · Force · Impulse · Torque / Moment / Couple · Momentum · Contents Angular momentum · Inertia · Moment of inertia · Reference frame · 1 Lagrange equations of the second kind Energy · Kinetic energy · Potential energy · 1.1 Kinetic energy relations Mechanical work · Virtual work · 2 Old Lagrange's equations D'Alembert's principle 3 Examples 3.1 Falling mass Core topics 3.2 Pendulum on a movable support Rigid body · Rigid body dynamics · 3.3 Two-body central force problem Euler's equations (rigid body dynamics) · 4 Hamilton's principle Motion · Newton's laws of motion · 5 Extensions of Lagrangian mechanics Newton's law of universal gravitation · 6 See also Equations of motion · 7 References Inertial frame of reference · 8 Further reading 9 External links Non-inertial reference frame · Rotating reference frame · Fictitious force · Lagrange equations of the second kind Linear motion · Mechanics of planar particle motion · The equations of motion in Lagrangian mechanics are the Lagrange equations , also Displacement (vector) · Relative velocity · known as the Euler–Lagrange equations . Below, we sketch out the derivation of the Friction · Simple harmonic motion · Lagrange equations of the second kind. Please note that in this context, V is used Harmonic oscillator · Vibration · Damping · rather than U for potential energy and T replaces K for kinetic energy. See the Damping ratio · Rotational motion · references for more detailed and more general derivations. Circular motion · Uniform circular motion · Start with D'Alembert's principle for the virtual work of applied forces, , and Non-uniform circular motion · inertial forces on a three dimensional accelerating system of n particles, i, whose Centripetal force · Centrifugal force · motion is consistent with its constraints, [5]:269 Centrifugal force (rotating reference frame) · Reactive centrifugal force · Coriolis force · Pendulum · Rotational speed · Angular acceleration · Angular velocity · Angular frequency · Angular displacement Scientists http://en.wikipedia.org/wiki/Lagrangian_mechanics 5/23/2011 Lagrangian mechanics - Wikipedia, the free encyclopedia Page 2 of 11 where Galileo Galilei · Isaac Newton · δW is the virtual work; Jeremiah Horrocks · Leonhard Euler · is the virtual displacement of the system, consistent with the constraints; Jean le Rond d'Alembert · Alexis Clairaut · mi are the masses of the particles in the system; Joseph Louis Lagrange · are the accelerations of the particles in the system; Pierre-Simon Laplace · together as products represent the time derivatives of the system momenta, aka. inertial forces; William Rowan Hamilton · i is an integer used to indicate (via subscript) a variable corresponding to a Siméon-Denis Poisson particular particle; and n is the number of particles under consideration. Break out the two terms: Assume that the following transformation equations from m independent generalized coordinates, q , hold: [5]:260 j where m (without a subscript) indicates the total number of generalized coordinates. An expression for the virtual displacement (differential), of the system for time-independent constraints is [5]:264 where j 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, Q :[5]:265 j δ Combining the equations for W, , and Qj yields the following result after pulling the sum out of the dot product in the second term: [5]:269 Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the kinetic energy leaves [5]:270 δ In the above equation, qj is arbitrary, though it is by definition consistent with the constraints. So the relation must hold term- wise: [5]:270 http://en.wikipedia.org/wiki/Lagrangian_mechanics 5/23/2011 Lagrangian mechanics - Wikipedia, the free encyclopedia Page 3 of 11 If the are conservative, they may be represented by a scalar potential field, V:[5]:266 & 270 The previous result may be easier to see by recognizing that V is a function of the , which are in turn functions of qj, and then applying the chain rule to the derivative of V with respect to qj. Recall the definition of the Lagrangian is [5]:270 Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows: This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to and time, and solely with respect to qj, adding the results and associating terms with the equations for and Qj. In a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the , Rayleigh suggests using a dissipation function, D, of the following form: [5]:271 where Cjk are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them If D is defined this way, then [5]:271 and Kinetic energy relations The kinetic energy, T, for the system of particles is defined by [5]:269 The partial derivative of T with respect to the time derivatives of the generalized coordinates, , is [5]:269 http://en.wikipedia.org/wiki/Lagrangian_mechanics 5/23/2011 Lagrangian mechanics - Wikipedia, the free encyclopedia Page 4 of 11 The previous result may be difficult to visualize. As a result of the product rule, the derivative of a general dot product is 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, and are equal to , which is why the factor of one half disappears. According to the chain rule and the coordinate transformation equations given above for , its time derivative, , is [5]:264 Together, the definition of and the total differential, , suggest that [5]:269 since and that in the sum, there is only one Substituting this relation back into the expression for the partial derivative of T gives [5]:269 Taking the time derivative gives [5]:270 Using the chain rule on the last term gives [5]:270 From the expression for , one sees that also [5]:270 This allows simplification of the last term, [5]:270 [5]:270 The partial derivative of T with respect to the generalized coordinates, qj, is http://en.wikipedia.org/wiki/Lagrangian_mechanics 5/23/2011 Lagrangian mechanics - Wikipedia, the free encyclopedia Page 5 of 11 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: [5]:270 Old Lagrange's equations Consider a single particle with mass m and position vector , moving under an applied force, , which can be expressed as the gradient of a scalar potential energy function : Such a force is independent of third- or higher-order derivatives of , so Newton's second law forms a set of 3 second-order ordinary differential equations. Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom . An obvious set of variables is , the Cartesian components of and their time derivatives, at a given instant of time (i.e. position (x,y,z) and velocity (vx,vy,vz)). More generally, we can work with a set of generalized coordinates, qj, and their time derivatives, the generalized velocities, . The position vector, , is related to the generalized coordinates by some transformation equation : 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 The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system.
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