Lagrangian mechanics
Information about Lagrangian mechanics
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 which minimizes the action functional, a quantity which 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 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.
Start with D'Alembert's principle for the virtual work of applied forces,
, and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints:[1]
Break out the two terms:
Assume that the following transformation equations from m independent generalized coordinates,
, hold:[1]
An expression for the virtual displacement (differential),
, of the system is[1]
The applied forces may be expressed in the generalized coordinates as generalized forces,
,[1]
Combining the equations for
, , and yields the following result after pulling the sum out of the dot product in the second term:[1]
Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the potential energy leaves[1]
In the above equation,
is arbitrary, though it is—by definition—consistent with the constraints. So the relation must hold term-wise:[1]
If the
are conservative, they may be represented by a scalar potential field, 
:[1]
.
The definition of the Lagrangian is[1]
Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows:[1]
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:[1]
If is defined this way, then[1]
, for the system of particles is defined by[1]
The partial derivative of with respect to the time derivatives of the generalized coordinates, 
, is[1]
According to the chain rule and the coordinate transformation equations given above for
, it's time derivative, 
, is:[1]
Together, the definition of
and the total differential, 
, suggest that[1]
Substituting this relation back into the expression for the partial derivative of gives[1]
Taking the time derivative gives[1]
Using the chain rule on the last term gives[1]
From the expression for, one sees that[1]
This allows simplification of the last term,[1]
The partial derivative of with respect to the generalized coordinates, , is[1]
The last two equations may be combined to give an expression for the inertial forces in terms of the kinetic energy:[1]
, 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 
).
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 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.
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[Confusing — Please clarify], we obtain:
Rearranging Slightly:
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's second law.
The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N 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 appropriate generalized coordinates qi may be chosen to exploit symmetries in the system.
from which the solution
and the potential energy is 
, hence
, yielding the same result as earlier.
Now carrying out the differentiations gives for the support coordinate x
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.
, 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:
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 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.
In 1948, 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.
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 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 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.Start with D'Alembert's principle for the virtual work of applied forces,
, and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints:[1]
.
- : 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
Break out the two terms:
.
Assume that the following transformation equations from m independent generalized coordinates,
, hold:[1]
,
, ...
.
An expression for the virtual displacement (differential),
, of the system is[1]
.
The applied forces may be expressed in the generalized coordinates as generalized forces,
,[1]
.
Combining the equations for
, , and yields the following result after pulling the sum out of the dot product in the second term:[1]
.
Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the potential energy leaves[1]
.
In the above equation,
is arbitrary, though it is—by definition—consistent with the constraints. So the relation must hold term-wise:[1]
.
If the
are conservative, they may be represented by a scalar potential field, :[1]
.
The definition of the Lagrangian is[1]
.
Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows:[1]
.
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:[1]
.
- :
are constants and are not necessarily equal to the damping coefficients in a particular physical system
If
is defined this way, then[1]
and
.
Kinetic energy relations
The kinetic energy,, for the system of particles is defined by[1]
.
The partial derivative of
with respect to the time derivatives of the generalized coordinates, , is[1]
.[Confusing — Please clarify]
According to the chain rule and the coordinate transformation equations given above for
, it's time derivative, , is:[1]
.
Together, the definition of
and the total differential, , suggest that[1]
.[Confusing — Please clarify]
Substituting this relation back into the expression for the partial derivative of
gives[1]
.
Taking the time derivative gives[1]
.
Using the chain rule on the last term gives[1]
.
From the expression for
, one sees that[1]
.
This allows simplification of the last term,[1]
.
The partial derivative of
with respect to the generalized coordinates, , is[1]
.[Confusing — Please clarify]
The last two equations may be combined to give an expression for the inertial forces in terms of the kinetic energy:[1]
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 ).
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 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.
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[Confusing — Please clarify], we obtain:
Rearranging Slightly:
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's second law.
The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N 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 appropriate generalized coordinates qi 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 from which the solution
and the potential energy is , hence
.
, 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
)
Sketch of the situation with definition of the coordinates (click to enlarge)
;
.
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.
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 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 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, 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, 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
References
1. ^ Torby, Bruce (1984). "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4.
- 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
- 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
Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies.
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conservation of energy states that the total amount of energy in any closed system remains constant but can be recreated, although it may change forms, e.g. friction turns kinetic energy into thermal energy.
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Joseph Louis, comte de Lagrange
Joseph Louis Lagrange
Born January 25 1736
Turin, Italy
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Joseph Louis Lagrange
Born January 25 1736
Turin, Italy
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8th century - 9th century - 10th century
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885 886 887 - 888 - 889 890 891
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850s 860s 870s - 880s - 890s 900s 910s
885 886 887 - 888 - 889 890 891
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"Generalized coordinates are unspecified coordinates. By deriving equations of motion in terms of a general set of coordinates, the results found will be valid for any coordinate system that is ultimately specified.
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In mathematics, specifically in the calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform a problem from its weak formulation (variational form) into its strong formulation (differential equation).
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In physics, the action is a particular quantity in a physical system that can be used to describe its operation in an alternative manner to the usual differential equation approach. The action is not necessarily the same for different types of system.
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INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) is detecting some of the most energetic radiation that comes from space. It is the most sensitive gamma ray observatory ever launched.
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Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics.
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Analysis may refer to:
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Chemistry
- Analytical chemistry, to examine material samples to gain an understanding of their chemical composition
- Isotope analysis, the identification of isotopic signature, the distribution of certain stable isotopes and chemical
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Newton's laws of motion are three physical laws which provide relationships between the forces acting on a body and the motion of the body, first compiled by Sir Isaac Newton.
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The Euler-Lagrange equation or Lagrange's equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. It provides a way to solve for functions which extremize a given cost functional.
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D'Alembert's principle, also known as the Lagrange-D'Alembert principle, is a statement of the fundamental classical laws of motion. It is equivalent to Newton's second law. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert.
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Virtual work on a system is the work resulting from either virtual forces acting through a real displacement or real forces acting through a virtual displacement.
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Inertia is a property of matter by which it remains at rest or in uniform motion in the same straight line unless acted upon by some external force The principle of inertia is one of the fundamental principles of classical physics which are used to describe the motion of matter and
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"Generalized coordinates are unspecified coordinates. By deriving equations of motion in terms of a general set of coordinates, the results found will be valid for any coordinate system that is ultimately specified.
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"A virtual displacement is an assumed infinitesimal change of system coordinates occurring while time is held constant. It is called virtual rather than real since no actual displacement can take place without the passage of time.
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Generalized forces are defined via coordinate transformation of applied forces, , on a system of n particles, i. The concept finds use in Lagrangian mechanics, where it plays a conjugate role to generalized coordinates.
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A scalar potential is a fundamental concept in vector analysis and physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion with vector potential).
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Lagrangian, , of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics.
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Viscosity is a measure of the resistance of a fluid to deform under either shear stress or extensional stress. It is commonly perceived as "thickness", or resistance to flow.
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Lord Rayleigh
John William Strutt, 3rd Baron Rayleigh
Born 12 November 1842
Langford Grove, Maldon, Essex, UK
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John William Strutt, 3rd Baron Rayleigh
Born 12 November 1842
Langford Grove, Maldon, Essex, UK
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kinetic energy of an object is the extra energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its current velocity.
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A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as .
A variety of notations are used to denote the time derivative.
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A variety of notations are used to denote the time derivative.
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In calculus, the chain rule is a formula for the derivative of the composite of two functions.
In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
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In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
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Mass is a fundamental concept in physics, roughly corresponding to the intuitive idea of "how much matter there is in an object". Mass is a central concept of classical mechanics and related subjects, and there are several definitions of mass within the framework of relativistic
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A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary inertial frame of reference, referred to as a reference or location "point" that exists in 2 or 3 dimensional space.
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In physics, force is an action or agency that causes a body of mass m to accelerate. It may be experienced as a lift, a push, or a pull. The acceleration of the body is proportional to the vector sum of all forces acting on it (known as net force or resultant force).
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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Newton's laws of motion are three physical laws which provide relationships between the forces acting on a body and the motion of the body, first compiled by Sir Isaac Newton.
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