Everything about Lagrangian Mechanics totally explained
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 that minimizes the
action functional, a quantity that's 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 the 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 isn't 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,
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're a differential principle) as the basis for mechanics. However it isn't widely stated that Hamilton's principle is a variational principle only with
holonomic constraints, if we're 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've 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.
Further Information
Get more info on 'Lagrangian Mechanics'.
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