For a more modern approach, see Lagrangian.

**Lagrangian mechanics** is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of an object is derived by finding the path which minimizes the action which is the sum of the Lagrangian over time; this being the kinetic energy minus the potential energy.

This considerably simplifies many physical problems. For example, consider a bead on a hoop. If one were to calculate the motion of the bead using Newtonian mechanics, one would have a complicated set of equations which would take into account the forces that the hoop exerts on the bead at each moment.

The same problem using Lagrangian mechanics is much simpler. One looks at all the possible motions that the bead could take on the hoop and mathematically finds the one which minimizes the action. There are many fewer equations since one is not directly calculating the influence of the hoop on the bead at a given moment.

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Consider a single particle with mass *m* and position vector **r**. The applied force, **F**, can be expressed as the gradient of a scalar potential energy function *V*(**r**, *t*):

More generally, we can work with a set of *generalized coordinates* and their time derivatives, the *generalized velocities*: {*q*_{j}, *q*′_{j}}. **r** is related to the generalized coordinates by some *transformation equation*:

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 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 *q*_{i} may be chosen to exploit symmetries in the system.

The action, denoted by *S*, is the time integral of the Lagrangian:

*The system undergoes the trajectory between t*_{0}and t_{1}whose action has a stationary value.

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 either a maximum, saddle point, or minimum in the action.

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.

In 1948, Feynman invented the path integral formulation extending the Principle of Least Action to quantum mechanics. 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.