page_title: Lagrangian mechanics
section_title: Introduction
hierarchical_section_title: Lagrangian mechanics / Introduction
context_page_description: Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis 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, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.
  No new physics is necessarily introduced in applying Lagrangian mechanics compared to Newtonian mechanics. It is, however, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction; however, they must include constraint forces explicitly and are best suited to Cartesian coordinates.
context_section_description: Suppose there exists a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects (bead, pendulum bob, etc.) as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion (reaction force exerted by the wire on the bead, or tension in the pendulum rod). For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that completely characterize the possible motion of the particle. 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 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 r₁ = (x₁, y₁, z₁), r₂ = (x₂, y₂, z₂) 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 r = (x, y, z). 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.
  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 can be defined by
  where
  is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, and V is the potential energy of the system.
  Kinetic energy is the energy of the system's motion, and vₖ² = vₖ · vₖ is the magnitude squared of velocity, equivalent to the dot product of the velocity with itself. The kinetic energy is a function only of the velocities vₖ, not the positions rₖ nor time t, so T = T(v₁, v₂, ...).
  The potential energy of the system reflects the energy of interaction between the particles, i.e. how much energy any one particle will have due to all the others and other external influences. For conservative forces (e.g. Newtonian gravity), it is a function of the position vectors of the particles only, so V = V(r₁, r₂, ...). For those non-conservative forces which can be derived from an appropriate potential (e.g. electromagnetic potential), the velocities will appear also, V = V(r₁, r₂, ..., v₁, v₂, ...). If there is some external field or external driving force changing with time, the potential will change with time, so most generally V = V(r₁, r₂, ..., v₁, v₂, ..., t).
  The above form of L does not hold in relativistic Lagrangian mechanics, and must be replaced by a function consistent with special or general relativity. Also, for dissipative forces another function must be introduced alongside L.
  One or more of the particles may each be subject to one or more holonomic constraints; such a constra
caption: Simple pendulum. Since the rod is rigid, the position of the bob is constrained according to the equation f(x, y) = 0, the constraint force C is the tension in the rod. Again the non-constraint force N in this case is gravity.