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 fewer equations since one is not directly calculating the influence of the hoop on the bead at a given moment.
The equations of motion in Lagrangian mechanics are ''Lagrange's equations'', also known as '' Euler-Lagrange Equation s''. 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 . The applied Force , '''F''', can be expressed as the Gradient of a scalar potential energy function ''V''(, ''t''):
:
|
\delta q_i = 0.
However, this must be true for ''any'' set of generalized displacements δ''q''
i, so we must have
:
for ''each'' generalized coordinate δ''q''
i. We can further simplify this by noting that ''V'' is a function solely of and ''t'', and 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'', called the Lagrangian, 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 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 Equation s 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'':
The system undergoes the trajectory between t0 and t1 whose action has a stationary value.
By ''stationary'', we mean that the action does not vary to first-order for infinitesimal deformations of the trajectory, with the end-points (''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 '', 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 (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 .
- Goldstein, H. ''Classical Mechanics,'' second edition, pp.16 (Addison-Wesley, 1980)
- Moon, F. C. ''Applied Dynamics With Applications to Multibody and Mechatronic Systems'', pp. 103-168 (Wiley, 1998).
