Lagrangian mechanics is a reformulation 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.
Lagrange's Equations The equations of motion in Lagrangian mechanics are Lagrange's equations, also known as EulerLagrange equations. Below, we sketch out the derivation of Lagrange's equation from Newton's laws of motion. See the references for more detailed and more general derivations.
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):
Such a force is independent of third or higherorder derivatives of r, so Newton's Second Law forms a set of 3 secondorder 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 {r_{j}, r′_{j}  j = 1, 2, 3}, the Cartesian components of r and their time derivatives, at a given instant of time.
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:
Consider an arbitrary displacement δr of the particle. The work done by the applied force F is δW = F · δr. Using Newton's Second Law, we write:
\mathbf{F} \cdot \delta \mathbf{r} & = & m\mathbf{r} \cdot \delta \mathbf{r}\end{matrix}</math>
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,
\begin{matrix} \mathbf{F} \cdot \delta \mathbf{r} & = &  \nabla V \cdot \sum_i {\partial r \over \partial q_i} \delta q_i \\ \\ & = &  \sum_{i,j} {\partial V \over \partial r_j} {\partial r_j \over \partial q_i} \delta q_i \\ \\ & = &  \sum_i {\partial V \over \partial q_i} \delta q_i \\ \end{matrix}</math>
The right hand side is more difficult, but after some shuffling we obtain:
m \mathbf{r} \cdot \delta \mathbf{r}
However, this must be true for any set of generalized displacements δq_{i}, so we must have
for each generalized coordinate δq_{i}. 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, we obtain Lagrange's equations:
There is one Lagrange equation for each generalized coordinate q_{i}. 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.
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 EulerLagrange 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:
Let 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 the Lagrange's equations are equivalent to Hamilton's Principle:
By stationary, we mean that the action does not vary to firstorder for 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, 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.
References
Goldstein, H. Classical Mechanics, second edition, pp.16 (AddisonWesley, 1980)
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