Lagrangian mechanics is a re-formulation of
classical mechanicsthat combines conservation of momentumwith conservation of energy. It was introduced by Italian mathematician Lagrangein 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 variationsshows that solving Lagrange's equation is equivalent to finding the path that minimizes the action functional, a quantity that is the integralof the Lagrangianover 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 mechanicswould 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.
The equations of motion in Lagrangian mechanics are "Lagrange's equations", also known as "
Euler–Lagrange equations". 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.
D'Alembert's principlefor the virtual workof applied forces, , and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints: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] rp|269
:.:: is the virtual work:: is the virtual displacement of the system, consistent with the constraints:: are the masses of the particles in the system:: are the accelerations of the particles in the system:: together as products represent the time derivatives of the system momenta, aka. inertial forces:: is an integer used to indicate (via subscript) a variable corresponding to a particular particle:: is the number of particles under consideration
Break out the two terms:
:,:, ...:.:: (without a subscript) indicates the total number generalized coordinates
:.:: is an integer used to indicate (via subscript) a variable corresponding to a generalized coordinate
The previous result may be easier to see by recognizing that is a function of the , which are in turn functions of , and then applying the
chain ruleto the derivative of with respect to .
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 , adding the results and associating terms with the equations for and .
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, , of the following form:rp|271
:.:: are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them
Kinetic energy relations
The previous result may be difficult to visualize. As a result of the
product rule, the derivative of a general dot productis 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.
[ Remember that :. Also remember that in the sum, there is only one . ]
[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:rp|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 gradientof 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 ).
More generally, we can work with a set of
generalized coordinates, , 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 pendulumof 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.
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 coordinatesClarifyme|date=March 2008, we obtain:
Now, by performing an "integration by parts" 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
However, this must be true for "any" set of generalized displacements , so we must have
for "each" generalized coordinate . 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:
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 qi. 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'ssecond 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 ofthe 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 appropriate generalized coordinates "q"i may be chosen to exploit symmetries in the system.
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.
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 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 coordinate, which is "0" at the starting point. The kinetic energy is and the potential energy is , hence :. Now we find: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
Now carrying out the differentiations gives for the support coordinate "x":therefore::indicating the presence of a constant of motion. The other variable 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 prone to errors. By considering limit cases ( should give the equations of motion for a pendulum, should give the equations for a pendulum in a constantly accelerating system, etc.) the correctness of this system can be verified.
The action, denoted by , is the time integral of the Lagrangian::
Let "q0" and "q1" be the coordinates at respective initial and final times "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 t0 and t1 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, 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 Lawsas 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 holonomicconstraints, if we are dealing with nonholonomic systems then the variational principle should be replaced with one involving d'Alembertprinciple 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
Legendre transformationon 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)).
1948, Feynman invented the path integral formulationextending the principle of least actionto quantum mechanicsfor electronsand 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 principlein optics.
Lagrangian analysis(applications of Lagrangian mechanics)
Restricted three-body problem
* 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).
* 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]
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