Lagrangian mechanics
Classical mechanics  

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Core topics  
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Lagrangian mechanics is a reformulation of classical mechanics using the principle of stationary action (also called the principle of least action).^{[1]} Lagrangian mechanics applies to systems whether or not they conserve energy or momentum, and it provides conditions under which energy, momentum or both are conserved.^{[2]} It was introduced by the ItalianFrench mathematician JosephLouis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind,^{[3]} which treat constraints explicitly as extra equations, often using Lagrange multipliers;^{[4]}^{[5]} or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.^{[3]}^{[6]} The fundamental lemma of the calculus of variations shows that solving the Lagrange equations is equivalent to finding the path for which the action functional is stationary, 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 timevarying 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. Lagrangian mechanics ability to work in generalized coordinate systems makes it suitable for use in special relativity which coordinate systems change for different observers. It is used as the standard language of particle physics to express the standard model in terms of Lagrangians.
ContentsConceptual frameworkGeneralized coordinatesConcepts and terminologyFor one particle acted on by external forces, Newton's second law forms a set of 3 secondorder ordinary differential equations, one for each dimension. Therefore, the motion of the particle can be completely described by 6 independent variables: 3 initial position coordinates and 3 initial velocity coordinates. Given these, the general solutions to Newton's second law become particular solutions that determine the time evolution of the particle's behaviour after its initial state (t = 0). The most familiar set of variables for position r = (r_{1}, r_{2}, r_{3}) and velocity '"`UNIQpostMath00000002QINU`"' are Cartesian coordinates and their time derivatives (i.e. position (x, y, z) and velocity (v_{x}, v_{y}, v_{z}) components). Determining forces in terms of standard coordinates can be complicated, and usually requires much labour. An alternative and more efficient approach is to use only as many coordinates as are needed to define the position of the particle, at the same time incorporating the constraints on the system, and writing down kinetic and potential energies. In other words, to determine the number of degrees of freedom the particle has, i.e. the number of possible ways the system can move subject to the constraints (forces that prevent it moving in certain paths). Energies are much easier to write down and calculate than forces, since energy is a scalar while forces are vectors. These coordinates are generalized coordinates, denoted '"`UNIQpostMath00000003QINU`"', and there is one for each degree of freedom. Their corresponding time derivatives are the generalized velocities, '"`UNIQpostMath00000004QINU`"'. The number of degrees of freedom is usually not equal to the number of spatial dimensions: multibody systems in 3dimensional space (such as Barton's Pendulums, planets in the solar system, or atoms in molecules) can have many more degrees of freedom incorporating rotations as well as translations. This contrasts the number of spatial coordinates used with Newton's laws above. Mathematical formulationThe position vector r in a standard coordinate system (like Cartesian, spherical etc.), is related to the generalized coordinates by some transformation equation:
where there are as many q_{i} as needed (number of degrees of freedom in the system). Likewise for velocity and generalized velocities. For example, for a simple pendulum of length ℓ, there is the constraint of the pendulum bob's suspension (rod/wire/string etc.). The position r depends on the x and y coordinates at time t, that is, r(t)=(x(t),y(t)), however x and y are coupled to each other in a constraint equation (if x changes y must change, and vice versa). A logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, so we have r = (x(θ), y(θ)) = r(θ), in which θ = θ(t). Then the transformation equation would be
and so
which corresponds to the one degree of freedom the pendulum has. The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system. In general, from m independent generalized coordinates q_{j}, the following transformation equations hold for a system composed of n particles:^{[7]}^{:260}
where m indicates the total number of generalized coordinates. An expression for the virtual displacement (infinitesimal), δr_{i} of the system for timeindependent constraints or "velocitydependent constraints" is the same form as a total differential^{[7]}^{:264}
where j is an integer label corresponding to a generalized coordinate. The generalized coordinates form a discrete set of variables that define the configuration of a system. The continuum analogue for defining a field are field variables, say ϕ(r, t), which represents density function varying with position and time. D'Alembert's principle and generalized forcesD'Alembert's principle introduces the concept of virtual work due to applied forces F_{i} and inertial forces, acting on a threedimensional accelerating system of n particles whose motion is consistent with its constraints,^{[7]}^{:269} Mathematically, the virtual work δW done on a particle of mass m_{i} through a virtual displacement δr_{i} (consistent with the constraints) is:
where a_{i} are the accelerations of the particles in the system and i = 1, 2,...,n simply labels the particles. In terms of generalized coordinates
this expression suggests that the applied forces may be expressed as generalized forces, Q_{j}. Dividing by δq_{j} gives the definition of a generalized force:^{[7]}^{:265}
If the forces F_{i} are conservative, there is a scalar potential field V in which the gradient of V is the force:^{[7]}^{:266 & 270}
i.e. generalized forces can be reduced to a potential gradient in terms of generalized coordinates. The previous result may be easier to see by recognizing that V is a function of the r_{i}, which are in turn functions of q_{j}, and then applying the chain rule to the derivative of '"`UNIQpostMath0000000EQINU`"' with respect to q_{j}. Kinetic energy relationsThe kinetic energy, T, for the system of particles is defined by^{[7]}^{:269}
The partial derivatives of T with respect to the generalized coordinates q_{j} and generalized velocities '"`UNIQpostMath00000010QINU`"' are:^{[7]}^{:269}
Because '"`UNIQpostMath00000013QINU`"' and '"`UNIQpostMath00000014QINU`"' are independent variables:
Then:
The total time derivative of this equation is
resulting in:
Newton's laws are contained in it, yet there is no need to find the constraint forces because virtual work and generalized coordinates (which account for constraints) are used. This equation in itself is not actually used in practice, but is a step towards deriving Lagrange's equations (see below).^{[8]} Lagrangian and actionThe core element of Lagrangian mechanics is the Lagrangian function, which summarizes the dynamics of the entire system in a very simple expression. The physics of analyzing a system is reduced to choosing the most convenient set of generalized coordinates, determining the kinetic and potential energies of the constituents of the system, then writing down the equation for the Lagrangian to use in Lagrange's equations. It is defined by ^{[9]}
where T is the total kinetic energy and V is the total potential energy of the system. The next fundamental element is the action '"`UNIQpostMath0000001AQINU`"', defined as the time integral of the Lagrangian:^{[8]}
This also contains the dynamics of the system, and has deep theoretical implications (discussed below). Technically, the action is a functional, that is, it is a function that maps the full Lagrangian function for all times between t_{1} and t_{2} to a scalar value for the action. Its dimensions are the same as angular momentum. In classical field theory, the physical system is not a set of discrete particles, but rather a continuous field defined over a region of 3d space. Associated with the field is a Lagrangian density '"`UNIQpostMath0000001CQINU`"' defined in terms of the field and its derivatives at a location '"`UNIQpostMath0000001DQINU`"'. The total Lagrangian is then the integral of the Lagrangian density over 3d space (see volume integral):
where d^{3}r is a 3d differential volume element, must be used instead. The action becomes an integral over space and time:
Hamilton's principle of stationary actionLet q_{0} and q_{1} be the coordinates at respective initial and final times t_{0} and t_{1}. Using the calculus of variations, it can be shown that Lagrange's equations are equivalent to Hamilton's principle:
By stationary, we mean that the action does not vary to firstorder from infinitesimal deformations of the trajectory, with the endpoints (q_{0}, t_{0}) and (q_{1},t_{1}) 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 the action functional need only be stationary, not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of 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. Lagrange equations of the first kindLagrange introduced an analytical method for finding stationary points using the method of Lagrange multipliers, and also applied it to mechanics. For a system subject to the (holonomic) constraint equation on the generalized coordinates:
where A is a constant, then Lagrange's equations of the first kind are:
where λ is the Lagrange multiplier. By analogy with the mathematical procedure, we can write:
where
denotes the variational derivative. For e constraint equations F_{1}, F_{2},..., F_{e}, there is a Lagrange multiplier for each constraint equation, and Lagrange's equations of the first kind generalize to:
This procedure does increase the number of equations, but there are enough to solve for all of the multipliers. The number of equations generated is the number of constraint equations plus the number of coordinates, i.e. e + m. The advantage of the method is that (potentially complicated) substitution and elimination of variables linked by constraint equations can be bypassed. There is a connection between the constraint equations F_{j} and the constraint forces N_{j} acting in the conservative system (forces are conservative):
which is derived below.
Lagrange equations of the second kindEuler–Lagrange equationsFor any system with m degrees of freedom, the Lagrange equations include m generalized coordinates and m generalized velocities. Below, we sketch out the derivation of the Lagrange equations of the second kind. 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. The equations of motion in Lagrangian mechanics are the Lagrange equations of the second kind, also known as the Euler–Lagrange equations:^{[8]}^{[10]}
where j = 1, 2,...m represents the jth degree of freedom, q_{j} are the generalized coordinates, and '"`UNIQpostMath0000002FQINU`"' are the generalized velocities. Although the mathematics required for Lagrange's equations appears significantly more complicated than Newton's laws, this does point to deeper insights into classical mechanics than Newton's laws alone: in particular, symmetry and conservation. In practice it's often easier to solve a problem using the Lagrange equations than Newton's laws, because the minimum generalized coordinates q_{i} can be chosen by convenience to exploit symmetries in the system, and constraint forces are incorporated into the geometry of the problem. There is one Lagrange equation for each generalized coordinate q_{i}. For a system of many particles, each particle can have different numbers of degrees of freedom from the others. In each of the Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy. Derivation of Lagrange's equationsHamilton's principleThe Euler–Lagrange equations follow directly from Hamilton's principle, and are mathematically equivalent. From the calculus of variations, any functional of the form:
leads to the general Euler–Lagrange equation for stationary value of J. (see main article for derivation):
Then making the replacements:
yields the Lagrange equations for mechanics. Since mathematically Hamilton's equations can be derived from Lagrange's equations (by a Legendre transformation) and Lagrange's equations can be derived from Newton's laws, all of which are equivalent and summarize classical mechanics, this means classical mechanics is fundamentally ruled by a variation principle (Hamilton's principle above). Generalized forcesFor a conservative system, since the potential field is only a function of position, not velocity, Lagrange's equations also follow directly from the equation of motion above:
simplifying to
This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to '"`UNIQpostMath00000035QINU`"' and time, and solely with respect to q_{j}, adding the results and associating terms with the equations for F_{i} and Q_{j}. Newton's lawsAs the following derivation shows, no new physics is introduced, so the Lagrange equations can describe the dynamics of a classical system equivalently as Newton's laws.
When q_{i} = r_{i} (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. Dissipation functionIn a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the F_{i}, Rayleigh suggests using a dissipation function, D, of the following form:^{[7]}^{:271}
where C_{jk} 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^{[7]}^{:271}
and
ExamplesIn 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 that is hard to solve with Newton's laws. Falling massConsider 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 '"`UNIQpostMath00000049QINU`"' from which the solution
follows (by taking the antiderivative of the antiderivative, and choosing the origin as the starting point). This result can also be derived through the Lagrangian formalism. Take x to be the coordinate, which is 0 at the starting point. The kinetic energy is T = ^{1}⁄_{2}mv^{2} and the potential energy is V = −mgx; hence,
Then
which can be rewritten as '"`UNIQpostMath0000004DQINU`"', yielding the same result as earlier. Pendulum on a movable supportConsider 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 xdirection. 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
The Lagrangian is therefore
Now carrying out the differentiations gives for the support coordinate x
therefore:
indicating the presence of a constant of motion. Performing the same procedure for the variable '"`UNIQpostMath00000053QINU`"' 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 more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, '"`UNIQpostMath00000056QINU`"' should give the equations of motion for a pendulum that is at rest in some inertial frame, while '"`UNIQpostMath00000057QINU`"' should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively. Twobody central force problemThe basic problem is that of two bodies in orbit about each other attracted by a central force. The Jacobi coordinates are introduced; namely, the location of the center of mass R and the separation of the bodies r (the relative position). The Lagrangian is then^{[11]}^{[12]}
where M is the total mass, μ is the reduced mass, and U the potential of the radial force. The Lagrangian is divided into a centerofmass 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 corotating 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:^{[13]}
which is the equation of motion for a onedimensional problem in which a particle of mass μ is subjected to the inward central force −dU/dr and a second outward force, called in this context the centrifugal force:
Of course, if one remains entirely within the onedimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general twodimensional problem from which the onedimensional problem originated. If one arrives at this equation using Newtonian mechanics in a corotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation 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:^{[14]} "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, 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.^{[15]} 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" will be omitted frequently." 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 symmetry of the system.^{[16]} Extensions of Lagrangian mechanicsThe Hamiltonian, denoted by H, is obtained by performing a Legendre transformation 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 quantum mechanics (see Hamiltonian (quantum mechanics)). In 1948, Feynman discovered 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. Dissipation (i.e. nonconservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom; see.^{[17]}^{[18]}^{[19]}^{[20]} See also
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