Lagrangian mechanics


In physics, Lagrangian mechanics is an alternate formulation of classical mechanics founded on the d'Alembert principle of virtual work. It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 culminating in his 1788 grand opus, Mécanique analytique. Lagrange's approach greatly simplifies the analysis of many problems in mechanics, and it had crucial influence on other branches of physics, including relativity and quantum field theory.
Lagrangian mechanics describes a mechanical system as a pair consisting of a configuration space M and a smooth function within that space called a Lagrangian. For many systems,, where T and V are the kinetic and potential energy of the system, respectively.
The stationary action principle requires that the action functional of the system derived from L must remain at a stationary point throughout the time evolution of the system. This constraint allows the calculation of the equations of motion of the system using Lagrange's equations.

Introduction

Newton's laws and the concept of forces are the usual starting point for teaching about mechanical systems. This method works well for many problems, but for others the approach is
nightmarishly complicated. For example, in calculation of the motion of a torus rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the angular velocity of the torus and the motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations. Lagrangian mechanics adopts energy rather than force as its basic ingredient, leading to more abstract equations capable of tackling more complex problems.
Particularly, Lagrange's approach was to set up independent generalized coordinates for the position and speed of every object, which allows the writing down of a general form of the Lagrangian and summing this over all possible paths of motion of the particles yielded a formula for the 'action', which he minimized to give a generalized set of equations. This summed quantity is minimized along the path that the particle actually takes. This choice eliminates the need for the constraint force to enter into the resultant generalized system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m1, m2,..., mN, each particle has a position vector, denoted r1, r2,..., rN. Cartesian coordinates are often sufficient, so, and so on. In three-dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all specific points in space to locate the particles; a general point in space is written. The velocity of each particle is how fast the particle moves along its path of motion, and is the time derivative of its position, thus
In Newtonian mechanics, the equations of motion are given by Newton's laws. The second law "net force equals mass times acceleration",
applies to each particle. For an N-particle system in 3 dimensions, there are 3N second-order ordinary differential equations in the positions of the particles to solve for.

Lagrangian

Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has units of energy, but no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian. It is nevertheless possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles in the absence of an electromagnetic field is given by
where
is the total kinetic energy of the system, equaling the sum Σ of the kinetic energies of the particles. Each particle labeled has mass and is the magnitude squared of its velocity, equivalent to the dot product of the velocity with itself.
Kinetic energy is the energy of the system's motion and is a function only of the velocities vk, not the positions rk, nor time t, so
V, the potential energy of the system, reflects the energy of interaction between the particles, i.e. how much energy any one particle has due to all the others, together with any external influences. For conservative forces, it is a function of the position vectors of the particles only, so For those non-conservative forces which can be derived from an appropriate potential, the velocities will appear also, If there is some external field or external driving force changing with time, the potential changes with time, so most generally

Mathematical formulation (for finite particle systems)

An equivalent but more mathematically formal definition of the Lagrangian is as follows.
For a system of N particles in three-dimensional space, the configuration space of the system is a smooth manifold, where each configuration specifies the spatial positions of each of the particles at a given instant of time, and the manifold is composed of all the configurations that are allowed by the constraints on the system.
The Lagrangian is a smooth function:
where is the tangent bundle of the configuration space. That is, each element in represents both the positions and the velocities of the particles, and can be written as a tuple with and specifying a position and a velocity of the i'th particle respectively. The time dependence allows for the Lagrangian to describe time-dependent forces or potentials.
A trajectory of the system is a smooth function
describing the evolution of the configuration over time. Its velocity is the time derivative of, and the pair is thus an element of the bundle for any. The action functional of the trajectory can therefore be defined as the integral of the Lagrangian along the path:
The laws of motion in Lagrangian mechanics are derived from the postulate that among all trajectories between two given configurations, the actual one that will be taken by the system must be a critical point of the action functional. This leads to the Euler–Lagrange equations.

Equations of motion

For a system of particles with masses, the kinetic energy is:
where is the velocity of particle i.
The potential energy depends only on the configuration, and typically arises from conservative forces.
The standard Lagrangian is given by the difference:
This formulation covers both conservative and time-dependent systems and forms the basis for generalizations to continuous systems, constrained systems, and systems with curved configuration spaces.
If T or V or both depend explicitly on time due to time-varying constraints or external influences, the Lagrangian is explicitly time-dependent. If neither the potential nor the kinetic energy depend on time, then the Lagrangian is explicitly independent of time. In either case, the Lagrangian always has implicit time dependence through the generalized coordinates.
With these definitions, Lagrange's equations are
where k = 1, 2,..., N labels the particles, there is a Lagrange multiplier λi for each constraint equation fi, and
are each shorthands for a vector of partial derivatives with respect to the indicated variables. Each overdot is a shorthand for a time derivative. This procedure does increase the number of equations to solve compared to Newton's laws, from 3N to, because there are 3N coupled second-order differential equations in the position coordinates and multipliers, plus C constraint equations. However, when solved alongside the position coordinates of the particles, the multipliers can yield information about the constraint forces. The coordinates do not need to be eliminated by solving the constraint equations.
In the Lagrangian, the position coordinates and velocity components are all independent variables, and derivatives of the Lagrangian are taken with respect to these separately according to the usual differentiation rules.
In each constraint equation, one coordinate is redundant because it is determined from the other coordinates. The number of independent coordinates is therefore. We can transform each position vector to a common set of n generalized coordinates, conveniently written as an n-tuple, by expressing each position vector, and hence the position coordinates, as functions of the generalized coordinates and time:
The vector q is a point in the configuration space of the system. The time derivatives of the generalized coordinates are called the generalized velocities, and for each particle the transformation of its velocity vector, the total derivative of its position with respect to time, is
Given this vk, the kinetic energy in generalized coordinates depends on the generalized velocities, generalized coordinates, and time if the position vectors depend explicitly on time due to time-varying constraints, so
With these definitions, the Euler–Lagrange equations,
are mathematical results from the calculus of variations, which can also be used in mechanics. Substituting in the Lagrangian gives the equations of motion of the system. The number of equations has decreased compared to Newtonian mechanics, from 3N to coupled second-order differential equations in the generalized coordinates. These equations do not include constraint forces at all, only non-constraint forces need to be accounted for.
Although the equations of motion include partial derivatives, the results of the partial derivatives are still ordinary differential equations in the position coordinates of the particles. The total time derivative denoted d/dt often involves implicit differentiation. Both equations are linear in the Lagrangian, but generally are nonlinear coupled equations in the coordinates.