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Lagrangian mechanics From Wikipedia, the free encyclopedia

Lagrangian mechanics is a re-formulation of that combines Classical mechanics conservation of with conservation of . It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the of a system of is derived by solving History of classical mechanics · the Lagrange 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 · / · · [1] incorporate the constraints directly by judicious choice of . · · [4] The fundamental lemma of the shows that solving the · Lagrange equations is equivalent to finding the path for which the is stationary, a quantity that is the integral of the Lagrangian over . 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 , calculation of the of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint required to  : 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  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 · Time · · · · influence of the groove on the bead at a given . · · Force · · / Moment / · Momentum · Contents · · · Reference frame ·  1 Lagrange equations of the second kind Energy · · ·  1.1 Kinetic energy relations Mechanical · ·  2 Old Lagrange's equations D'Alembert's principle  3 Examples  3.1 Falling mass Core topics  3.2 on a movable support · ·  3.3 Two-body 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 ·  7 References Inertial ·  8 Further reading  9 External links Non-inertial reference frame · · · Lagrange equations of the second kind · Mechanics of planar particle motion · The equations of motion in Lagrangian mechanics are the Lagrange equations , also (vector) · · known as the Euler–Lagrange equations . Below, we sketch out the derivation of the · · Lagrange equations of the second kind. Please note that in this context, V is used · · · 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. · Uniform circular motion · Start with D'Alembert's principle for the virtual work of applied , , and Non-uniform circular motion · inertial forces on a three dimensional accelerating system of n particles, i, whose · · motion is consistent with its constraints, [5]:269 Centrifugal force (rotating reference frame) · Reactive centrifugal force · · Pendulum · · · · · Scientists

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where · · δW is the virtual work; · · is the virtual displacement of the system, consistent with the constraints; Jean le Rond d'Alembert · ·

mi are the of the particles in the system; Joseph Louis Lagrange · are the of the particles in the system; Pierre-Simon Laplace · together as products represent the time of the system momenta, aka. inertial forces; · 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 , Q :[5]:265 j

δ Combining the equations for W, , and Qj yields the following result after pulling the sum out of the in the second term: [5]:269

Substituting in the result from the kinetic energy relations to change the inertial forces into a of the kinetic energy leaves [5]:270

δ In the above , qj is arbitrary, though it is by definition consistent with the constraints. So the relation must hold term- wise: [5]:270

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If the are conservative, they may be represented by a potential , V:[5]:266 & 270

The previous result may be easier to see by recognizing that V is a function of the , which are in functions of qj, and then applying the to the of V with respect to qj.

Recall the definition of the Lagrangian is [5]:270

Since the potential field is only a function of , 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 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 of T with respect to the time derivatives of the generalized coordinates, , is [5]:269

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The previous result may be difficult to visualize. As a result of the , the derivative of a general dot product is

This general result may be seen by briefly stepping into a Cartesian , 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 , , 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

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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 of a 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 , . 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 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.

Consider an arbitrary displacement of the particle. The work done by the applied force is . Using Newton's second law, we write:

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,

On the right hand side, carrying out a change of coordinates to generalized coordinates, we obtain:

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Rearranging slightly:

Now, by performing an "" transformation, with respect to t:

Recognizing that and , we obtain:

Now, by changing the order of differentiation, we obtain:

Finally, we change the order of summation:

Which is equivalent to:

where is the kinetic energy of the particle. Our equation for the work done becomes

δ However, this must be true for any set of generalized displacements qi, so we must have

δ for each generalized coordinate qi. 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:

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Inserting this into the preceding equation and substituting L = T − V, called the Lagrangian, we obtain Lagrange's equations:

There is one Lagrange equation for each generalized coordinate q i. When qi = ri (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 equations than Newton's laws. This is because not only may more appropriate generalized coordinates qi be chosen to exploit symmetries in the system, but constraint forces are replaced with simpler relations. 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 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 = mg is exerted on the mass (assuming g constant during the motion). Filling in the force in Newton's law, we find from which the solution

follows (choosing the origin at the starting point). This result can also be derived through the Lagrange formalism. Take x to be the 1 2 coordinate, which is 0 at the starting point. The kinetic energy is T = ⁄2mv and the potential energy is V = −mgx ; hence,

.

Then

which can be rewritten as , 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

and the potential energy of the system is

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The Lagrangian is therefore

Sketch of the situation with definition of the coordinates (click to enlarge)

Now carrying out the differentiations gives for the support coordinate x

therefore:

indicating the presence of a . Performing the same procedure for the variable θ yields:

therefore

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 more prone to errors. By considering limit cases (for example, should give the equations of motion for a pendulum which is at rest in some inertial frame, while should give the equations for a pendulum in a constantly accelerating system, etc.) the correctness of this system can be verified. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.

Two-body central force problem

The basic problem is that of two bodies in about each other attracted by a central force. The Jacobi coordinates are introduced; namely, the location of the R and the separation of the bodies r (the relative position). The Lagrangian is then [6][7]

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where M is the total mass, µ is the , and U the potential of the radial force. The Lagrangian is divided into a center-of- mass term and a relative motion term. The R equation from the Euler-Lagrange system is simply:

resulting in simple motion of the center of mass in a straight line at constant velocity. The relative motion is expressed in polar coordinates ( r, θ):

which does not depend upon θ, therefore an ignorable coordinate. The Lagrange equation for θ is then:

where ℓ is the conserved angular momentum. The Lagrange equation for r is:

or:

This equation is identical to the radial equation obtained using Newton's laws in a co-rotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. If the angular velocity is replaced by its value in terms of the angular momentum,

the radial equation becomes: [8]

which is the equation of motion for a one-dimensional problem in which a particle of mass µ is subjected to the inward central force −dU/d r and a second outward force, called in this context the centrifugal force:

Of course, if one remains entirely within the one-dimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general two-dimensional problem from which the one-dimensional problem originated.

If one arrives at this equation using Newtonian mechanics in a co-rotating frame, the interpretation is evident as the centrifugal force in that frame due to the of the frame itself. If one arrives at this equation directly by using the generalized coordinates ( r, θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of using polar coordinates . As Hildebrand says: [9] "Since such quantities are not true physical forces, they are often called inertia forces . Their presence or absence depends, not upon the particular problem at hand,

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but upon the coordinate system chosen ." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the centripetal force for a curved motion.

This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates. [10] Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an inertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as generalized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal always with generalized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" frequently will be omitted."

It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the of the system. [11] Hamilton's principle

The action, denoted by , is the time integral of the Lagrangian:

Let q0 and q1 be the coordinates at respective initial and final t0 and t1. 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 0 and t 1 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 ( q0, t0) and ( q1,t1) fixed. Hamilton's principle can be written as:

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, with any variation h of the functional giving an increase in the functional integral of the action. This is not, as is frequently misstated, required to be a maximum or a minimum of the action functional.

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 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, denoted by H, is obtained by performing a on the Lagrangian, which introduces new variables, canonically conjugate to the original variables. This doubles the number of variables, but makes differential equations first order. The Hamiltonian is the basis for an alternative formulation of classical mechanics known as Hamiltonian mechanics. It is a particularly ubiquitous quantity in (see Hamiltonian (quantum mechanics)).

In 1948, Feynman discovered the path integral formulation extending the principle of least action to quantum mechanics for and . 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 leading to it. In the classical regime, the path integral formulation cleanly reproduces Hamilton's principle, and Fermat's principle in . See also

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   Generalized coordinates  Hamiltonian mechanics  Lagrangian analysis (applications of Lagrangian mechanics)  Nielsen form  Restricted three-body problem  Lagrangian Point  References

1. ^ a b R. Dvorak, Florian Freistetter (2005). "§ 3.2 Lagrange equations of the first kind". Chaos and stability in planetary systems . Birkhäuser. p. 24. ISBN 3540282084. http://books.google.com/books?id=shYNuW0B0fsC&pg=PA24. 2. ^ H Haken (2006). Information and self-organization (3rd ed.). Springer. p. 61. ISBN 3540330216. http://books.google.com/books? id=tAfj4-xzyGwC&pg=PA61. 3. ^ Cornelius Lanczos (1986). "II §5 Auxiliary conditions: the Lagrangian λ-method". The variational principles of mechanics (Reprint of University of Toronto 1970 4th ed.). Courier Dover. p. 43. ISBN 0486650677. http://books.google.com/books? id=ZWoYYr8wk2IC&pg=PA43. 4. ^ Henry Zatzkis (1960). "§1.4 Lagrange equations of the second kind". In DH Menzel. Fundamental formulas of physics . 1 (2nd ed.). Courier Dover. p. 160. ISBN 0486605957. http://books.google.com/books?id=QgswE2BicW4C&pg=PA160. 5. ^ a b c d e f g h i j k l m n o p q r s t u v Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for . HRW Series in Mechanical . United States of America: CBS College Publishing. ISBN 0-03-063366-4. 6. ^ John Robert Taylor (2005). Classical mechanics . University Science Books. p. 297. ISBN 189138922X. http://books.google.com/books?id=P1kCtNr-pJsC&pg=PA297. 7. ^ The Lagrangian also can be written explicitly for a rotating frame. See Thanu Padmanabhan (2000). "§2.3.2 Motion in a rotating frame" . Theoretical Astrophysics: Astrophysical processes (3rd ed.). Cambridge University Press. p. 48. ISBN 0521566320. http://books.google.com/books?id=ZzJicsTIrAAC&pg=PA48. 8. ^ Louis N. Hand, Janet D. Finch (1998). Analytical mechanics . Cambridge University Press. pp. 140–141. ISBN 0521575729. http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA141. 9. ^ Francis Begnaud Hildebrand (1992). Methods of (Reprint of Prentice-Hall 1965 2nd ed.). Courier Dover. p. 156. ISBN 0486670023. http://books.google.com/books?id=17EZkWPz_eQC&pg=PA156. 10. ^ For example, see Michail Zak, Joseph P. Zbilut, Ronald E. Meyers (1997). From instability to intelligence . Springer. p. 202. ISBN 3540630554. http://books.google.com/books?id=tHdDL0GCA70C&pg=PA202. for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in Ahmed A. Shabana (2008). Computational continuum mechanics . Cambridge University Press. pp. 118–119. ISBN 0521885698. http://books.google.com/books?id=RJbPQPrS6VsC&pg=PA118. 11. ^ Terry Gannon (2006). Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics . Cambridge University Press. p. 267. ISBN 0521835313. http://books.google.com/books?id=ehrUt21SnsoC&pg=RA3-PA267.

 Goldstein, H. Classical Mechanics, second edition, pp. 16 (Addison-Wesley, 1980)  , F. C. Applied Dynamics With Applications to Multibody and Mechatronic Systems , pp. 103–168 (Wiley, 1998). Further reading

 Landau, L.D. and Lifshitz, E.M. Mechanics , Pergamon Press.  Gupta, Kiran Chandra, Classical mechanics of particles and rigid bodies (Wiley, 1988). External links

 Tong, David, Classical Dynamics Cambridge lecture notes  Principle of least action interactive Excellent interactive explanation/webpage  Aerospace dynamics lecture notes on Lagrangian mechanics  Aerospace dynamics lecture notes on Rayleigh dissipation function  Introduction to Lagrangian Mechanics  Sydney Grammar School Academic Extension notes Retrieved from "http://en.wikipedia.org/wiki/Lagrangian_mechanics" Categories: Lagrangian mechanics

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