Lagrangian mechanics

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, a quantity which is the integral of the Lagrangian over time. The Lagrangian for classical mechanics is taken to be the difference between the kinetic energy and 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 fewer equations since one is not directly calculating the influence of the hoop on the bead at a given moment. One of the key advantages of Lagrangian mechanics is that no vector quantities are involved in the calculations, which simplifies problems a great deal.

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 from Newton's laws of motion. 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.

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

${\mathbf {F}}=-{\mathbf {\nabla }}V.$

Such a force is independent of third- or higher-order derivatives of ${\mathbf {r}}$, so Newton's second law forms a set of 3 second-order 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 $\{{\mathbf {r}}_{j},{\dot {\mathbf {r}}}_{j}|j=1,2,3\}$, the Cartesian components of ${\mathbf {r}}$ and their time derivatives, at a given instant of time (i.e. position (x,y,z) and velocity $(v_{x},v_{y},v_{z})$).

More generally, we can work with a set of generalized coordinates, $q_{j}$, and their time derivatives, the generalized velocities, ${\dot {q_{j}}}$. The position vector, ${\mathbf {r}}$, is related to the generalized coordinates by some transformation equation:

${\mathbf {r}}={\mathbf {r}}(q_{i},q_{j},q_{k},t).$

For example, for a simple pendulum of length l, a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be

${\mathbf {r}}(\theta ,{\dot {\theta }},t)=(l\sin \theta ,l\cos \theta )$.

The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system.

Consider an arbitrary displacement $\delta {\mathbf {r}}$ of the particle. The work done by the applied force ${\mathbf {F}}$ is $W={\mathbf {F}}\cdot \delta {\mathbf {r}}$. Using Newton's second law, we write:

${\begin{matrix}{\mathbf {F}}\cdot \delta {\mathbf {r}}=m{\ddot {\mathbf {r}}}\cdot \delta {\mathbf {r}}.\end{matrix}}$

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,

${\begin{matrix}{\mathbf {F}}\cdot {\mathbf {\delta }}{\mathbf {r}}&=&-{\mathbf {\nabla }}V\cdot \displaystyle \sum _{i}{\partial {\mathbf {r}} \over \partial q_{i}}\delta q_{i}\\\\&=&-\displaystyle \sum _{i,j}{\partial V \over \partial r_{j}}{\partial r_{j} \over \partial q_{i}}\delta q_{i}\\\\&=&-\displaystyle \sum _{i}{\partial V \over \partial q_{i}}\delta q_{i}.\\\end{matrix}}$

The right hand side is more difficult, but after some shuffling we obtain:

$m{\ddot {\mathbf {r}}}\cdot \delta {\mathbf {r}}=\sum _{i}\left[{\mathrm {d} \over \mathrm {d} t}{\partial T \over \partial {\dot {q_{i}}}}-{\partial T \over \partial q_{i}}\right]\delta q_{i}$

where $T={\frac {1}{2}}m{\dot {\mathbf {r}}}\cdot {\dot {\mathbf {r}}}$ is the kinetic energy of the particle. Our equation for the work done becomes

$\sum _{i}\left[{\mathrm {d} \over \mathrm {d} t}{\partial {T} \over \partial {\dot {q_{i}}}}-{\partial {(T-V)} \over \partial q_{i}}\right]\delta q_{i}=0.$

However, this must be true for any set of generalized displacements $\delta q_{i}$, so we must have

$\left[{\mathrm {d} \over \mathrm {d} t}{\partial {T} \over \partial {\dot {q_{i}}}}-{\partial {(T-V)} \over \partial q_{i}}\right]=0$

for each generalized coordinate $\delta q_{i}$. 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:

${\mathrm {d} \over \mathrm {d} t}{\partial {V} \over \partial {\dot {q_{i}}}}=0$

Inserting this into the preceding equation and substituting L = T - V, called the Lagrangian, we obtain Lagrange's equations:

${\partial {L} \over \partial q_{i}}={\mathrm {d} \over \mathrm {d} t}{\partial {L} \over \partial {\dot {q_{i}}}}$

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 Euler-Lagrange equations than Newton's laws. This is because appropriate generalized coordinates q_i may be chosen to exploit symmetries in the system.

Hamilton's principle

The action, denoted by ${\mathcal {S}}$, is the time integral of the Lagrangian:

${\mathcal {S}}=\int L\,\mathrm {d} t.$

Let q₀ and q₁ be the coordinates at respective initial and final times t₀ and t₁. 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 (q₀, t₀) and (q₁,t₁) fixed. Hamilton's principle can be written as:

$\delta {\mathcal {S}}=0.\,\!$

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 are a differential principle) as the basis for mechanics. However it is not widely stated that Hamilton's principle is a variational principle only with holonomic constraints, if we are 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 have 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.

See also

• Restricted three-body problem
• Hamiltonian mechanics
• Functional derivative
• Nielsen form
• Canonical coordinates
• Generalized coordinates

References

• 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).

Further reading

• Landau, L.D. and Lifshitz, E.M. Mechanics, Pergamon Press.

External links

• Tong, David, Classical Dynamics Cambridge lecture notes
• Principle of least action interactive Excellent interactive explanation/webpage