User:Eugene M. Izhikevich/Proposed/Lagrangian mechanics
Lagrangian Mechanics is a fundamental invariant formulation of the fundamental laws of mechanics of particles and rigid bodies. It is closely related to original formulation of dynamics of Newton. It can be given a variational formulation.
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Newton's Laws
The most fundamental contribution to mechanics were Newton's three laws of motion for a particle (see Newton (1650}, Book I, Section 3, Propositions XI, XII, XIII)).
Assuming that the reference frame is inertial, they are as follows (see Bloch (2003)):
- Every particle continues in its state of rest or of uniform velocity in a straight line unless compelled to do otherwise by a force acting on it.
- The rate of change of linear momentum is equal to the impressed force and takes place in the direction of action of the force.
- To every action there is an equal and opposite reaction.
For a particle of constant mass \(m\ ,\) Newton's second law can be
written as:
\[m\ddot{\mathbf{x}}(t) = \mathbf{F}(t,\mathbf{x},\mathbf{\dot{x}}), \] where \(\mathbf{x} \in \mathbb{R}^3\) is the position vector of the particle and \(\mathbf{F}(t)\) is the impressed force vector, both measured with respect to an inertial frame.
Generalized Coordinates
Generalized coordinates are variables whose values uniquely specify the location in 3-space of each physical point of the system. A set of generalized coordinates is minimal in the sense that no set of fewer variables suffices to determine the locations of all points on the system. The number of variables in a set of generalized coordinates for a mechanical system is called the number of degrees of freedom of the system.
Generalized coordinates may be interpreted as coordinates for the system's configuration space, often denoted by \(Q\ .\) Generalized coordinates are sometimes called Lagrangian coordinates.
Suppose one has a system of \(N\) particles with respective Euclidean coordinates \((x_{3a-2},x_{3a-1},x_{3a})\ ,\) \( a = 1, \dots, N \ .\) The idea is to rewrite the dynamics in terms of the generalized coordinates \((q^1, q^2, \dots ,q^m)\ ,\) \( m \le n \ ,\) \( n = 3N \ .\) Suppose (see e.g. Pars, 1965) \( x_r = x_r(q^1,q^2,\dots, q^m), r = 1, \dots, n, \) so that \( \delta x_r=\sum_{i=1}^m \frac{\partial x_r}{\partial q ^i}\delta q ^i, \) where \(\delta x_r\) denotes a virtual displacement---an infinitesimal instantaneous change of the \(r\)-th Euclidean coordinate. If \( m < n \ ,\) the first set of equations above define position, or holonomic, constraints imposed on the system. The \(q^i\) are then a set of free variables, and the virtual displacements \(\delta x_r \) are by construction consistent with the imposed constraints.
One can extend this notion to that of a set of (linear in velocities) nonholonomic constraints
\( \sum_{k=1}^na_{ik}dq_k+a_{i0}dt=0 \)
where the virtual displacements are given by
\( \sum_{k=1}^na_{ik}\delta q_k=0\,. \)
The constraints are holonomic is they are integrable.
More generally we may have a set of nonholonomic constraints of the form
\( \phi_j(q,\dot{q},t)=0\,. \)
A constraint is said to be scleronomic if time does not appear explicitly in the equations of constraint. Otherwise it is said to rheonomic.
Note that in general one may define a Lagrangian system on a manifold defined by a set of holonomic constraints (the level set of \(m\) independent functions of the generalized coordinates.
If one assumes that the forces of constraint do zero virtual work (see the definition of virtual work below), then Newton's law in d'Alembertian form implies
\[ \sum_{i=1}^m \left\{\sum_{r=1}^n(m_r\ddot{x_r}-F_r) \frac{\partial x_r}{\partial q ^i}\right\}\delta q ^i \] = 0
where the \(F_r\) now include only forces other than the forces of constraint.
The condition that the forces of constraint do zero virtual work is typically a restriction to idealized rigid body systems with no friction or with friction only at the stationary contact point of rolling constraints. The use of the d'Alembertian form here is for convenience only and does not imply that Lagrangian mechanics depends logically on d'Alembert's principle.
The above expression can be written as
\[ \sum_{i=1}^m \left\{\frac{d}{dt} \frac{\partial T}{\partial \dot{ q} ^i}-\frac{\partial T}{\partial q^i}-Q_i \right\}\delta q^i = 0, \]
where \(T\) is the quadratic kinetic energy rewritten in terms of the variables \(q^i\) and \( Q_i=\sum_{r=1}^n F_r\frac{\partial x_r}{\partial q ^i} \) are the generalized forces.
In the case the \(Q_i\) arise from a potential, \(Q_i= - \frac{\partial V}{\partial q ^i}\) we define the Lagrangian to be \(L=T-V\) and (assuming the virtual displacements to be independent) obtain the Euler--Lagrange equations
\[ \frac{d}{dt}\frac{\partial L}{\partial\dot{ q} ^i}-\frac{\partial L}{\partial q^i} = 0. \]
It is interesting that Lagrange (Lagrange, 1788) did not derive the Lagrange equations of motion by variational methods, but he did so by requiring that simple force balance be covariant, that is, expressible in arbitrary generalized coordinates. For further information on the history of variational principles and the precise formulation of the principle of least action (see Bloch, 2003 and Marsden and Ratiu, 1999).
The history of variational principles and the so-called principle of least action is quite complicated, and we leave most of the details to other references. An interesting historical note is that the currently accepted notion of the "principle of least action" is regarded by some as being synonymous with "Hamilton's principle." Indeed, Feynman advocates this point of view. However, both historically and factually, Hamilton's principle and the principle of least action (which should really be called the principle of critical action are slightly different. Hamilton's principle involves varying the integral of the Lagrangian over all curves with fixed endpoint and fixed time. The principle of least action, on the other hand, involves variation of the quantity \( \int _a^b \sum_i\dot{q}^i\frac{\partial L}{\partial\dot{q}^i}\,dt \) called the reduced action or Maupertuis's action, over all curves with fixed energy. This principle assumes that the system is time-invariant, that is, the kinetic and potential energies of the system are time-independent, whereas Hamilton's principle is suitable for both time-dependent and time-independent Lagrangians.
The principle of critical action originated in Maupertuis's work, and because of that it is referred to as Maupertuis's principle in some of the literature. Maupertuis attempted to obtain for the corpuscular theory of light a theorem analogous to Fermat's principle of least time. Briefly put, the latter involves taking the variations of
\[ \int n\,dl, \]
where \(n\) is the refractive index over the path of the light. This gives rise to Snell's law. This law was discovered by the Dutch mathematician and geodesist Willebord Snel van Royen. (Because his name in Latin is "Snellius" the law is often called Snell's law. Maupertuis's principle was established by Euler in 1744 for the case of a single particle and in more generality by Lagrange in 1760.
It is curious that Lagrange dealt with the more difficult principle of critical action already in 1760, yet Hamilton's principle, which is simpler, came only much later in 1834 and 1835.
One can extend these ideas to the case of nonholonomic, or velocity,
constraints (see Bloch, 2003), but we do not do so here.
Hamilton's Principle
In this section we give a brief introduction to the Euler--Lagrange equations of motion for holonomic systems from the point of view of variational principles.
Let \(Q\) be the configuration space( he configuration space of a system is best thought of as a differentiable manifold, and generalized coordinates as a coordinate chart on this manifold) of a system with (generalized) coordinates \(q^i\ ,\) \(i = 1,\dots, m\ .\) We are given a real-valued function \(L(q^i,\dot{q}^i)\ ,\) called a Lagrangian. Often we choose \(L\) to be \(L = T - V\ ,\) where \(T(q,\dot q)\) is the kinetic energy of the system and \(V(q)\) is the potential energy.
Hamilton's principle singles out particular curves \(q(t)\) by the condition
\[ \delta \int_a^b L (q (t), \dot{ q} (t))\,dt = 0 , \]
To make this precise, let the variation of a trajectory \(q(\cdot)\) with fixed endpoints satisfying \(q(a)=q_a\) and \(q(b)=q_b\) be defined to be a smooth mapping
\[ (t,\epsilon)\mapsto q(t,\epsilon),\qquad a\le t\le b,\qquad \epsilon \in(-\delta,\delta)\subset \mathbb{R}, \] satisfying (i) \( q(t,0)=q(t) \ ,\) \(t\in[a,b]\ ,\) and (ii) \(q(a,\epsilon)=q_a\ ,\) \(q(b,\epsilon)=q_b\ .\)
Letting \(\delta q(t)=(\partial/\partial\epsilon)q(t,\epsilon)|_{\epsilon=0}\) be the virtual displacement corresponding to the variation of \(q\ ,\) we have \( \delta q(a)=\delta q(b)=0. \)
The precise meaning of Hamilton's principle is then the statement
\[ \frac{d }{d\epsilon} \int^a_b L (q (t, \epsilon), \dot{ q} (t, \epsilon))\,dt | _{\epsilon = 0} = 0 \] for all variations.
where the variation is over smooth curves in \(Q\) with fixed endpoints.
One can view Hamilton's principle in the following way: The quantity
\(\int_a^b L (q (t), \dot{ q} (t))\, dt\) is being extremized among
all curves with fixed endpoints; that is, the particular curve
\(q(t)\) that is sought is a critical point of the quantity
\(\int_a^b L (q (t), \dot{ q} (t))\,dt\) thought of as a function on
the space of curves with fixed endpoints. Examples show that the
quantity \(\int _a^b L \,dt\) being extremized need
not be minimized at a solution of the Euler--Lagrange equations, just
as critical points of functions need not be
minima. (Perhaps the simplest example of this comes up in
the study of geodesics on a sphere where geodesics that "go the
long way around the sphere" are critical points, but not minima. In
this example, \(L\) is just the kinetic energy of a point particle on
the sphere. See Gelfand and Fomin, 1963 for further information.)
A basic result of the calculus of variations is:
Hamilton's principle for a curve \(q(t)\) is equivalent to the condition that \(q(t)\) satisfy the Euler--Lagrange equations
\[ \frac{d}{d t} \frac{\partial L}{\partial \dot{ q} ^i}-\frac{\partial L}{\partial q^i} = 0. \]
The idea of the proof is as follows: Let \(\delta q\) be a virtual displacement of the curve \(q(t)\) corresponding to the variation \(q(t,\epsilon)\ .\) We may compute the variation of the integral in the definition corresponding to this variation of the trajectory \(q\) by differentiating with respect to \(\epsilon\) and using the chain rule. We obtain
\[ \int^a_b \left( \frac{\partial L}{\partial q^i}\delta q^i + \frac{\partial L}{\partial \dot{q}^i}{\delta \dot{q}^i}\right)\,dt = 0 , \]
where \(\delta\dot{q}^i= \frac{d}{dt}{\delta q^i}\ .\) Integrating by parts and using the boundary conditions \(\delta q^i=0\) at \(t=a\) and \(t=b\) yields the identity
\[ \int^a_b \left( -\frac{d}{d t} \frac{\partial L}{\partial \dot{ q} ^i} + \frac{\partial L}{\partial q^i} \right)\delta q^i\,dt = 0 . \]
Assuming a rich enough class of variations yields the result.
A critical aspect of the Euler--Lagrange equations is that they may
be regarded as a way to write Newton's second law in a manner that
makes sense in arbitrary curvilinear and even moving coordinate
systems. That is, the Euler--Lagrange formalism is \(covariant\ .\)
This is of enormous benefit, not only theoretically, but for
practical problems as well.
We note that the above derivation implies that the Lagrangian is not unique but is defined only up to an additive total time derivative.
It should be noted that Hamilton's principle above gives the correct differential equation of motion only in those systems in which the forces of constrain do zero virtual work. Otherwise, Hamilton's principle will still lead to the homogeneous Lagrange equations shown here, but those equations will not be correct without explicit inclusion of the frictional parts of the constraint forces as external forces (as outlined in the following section). The absence of the forces of constraint from the Lagrange equations in idealized frictionless systems is of great utility, since it allows the equations of motion to be solved without knowing these constraint forces in advance.
Conservation Laws and Noether's theorem
An important special case of the Lagrange equations occurs when one has cyclic variables. A variable \(q^i\) is cyclic if the Lagrangian is independent of that variable (but does depend on the corresponding velocity). In that case we obtain
\[ \frac{d}{dt}\frac{\partial L}{\partial\dot{ q} ^i} = 0. \]
Thus, the conjugate momentum \( p_i= \frac{\partial L}{\partial\dot{ q} ^i} \) corresponding to the cyclic variable\( q_i \) is conserved along the flow of the system. This is a special case of Noether's theorem which, loosely speaking, says that if Lagrangian is invariant under the action of a group (i.e. it has a natural symmetry), this leads to a conserved momentum.
The role of symmetries in Lagrangian mechanics is very important. A general treatment, which relied on the theory of group actions, may be found e.g. in Marsden and Ratiu (1999) and Bloch (2003).
Mechanical Systems with External Forces
In the presence of external forces, the equations of motion become
\[ \frac{d}{d t} \frac{\partial L}{\partial \dot{ q} ^i}-\frac{\partial L}{\partial q^i} = Q_i,\quad i = 1, \dots, m. \]
Here we regard the quantities \(Q _i\) as given by external agencies. Note that if these forces are derivable from a potential \(V\) in the sense that \(Q _i = - \partial V / \partial q ^i\ ,\) then these forces can be incorporated into the Lagrangian by adding \(-V\) to the Lagrangian. That is, this way of adding forces is consistent with the Euler--Lagrange equations themselves.
These equations can be derived from a variational-like principle, the Lagrange--d'Alembert principle for systems with external forces, as follows:
\[ \delta \int _a^b L ( q^i, \dot{q} ^i) \,dt + \int _a^b Q \cdot \delta q\,dt = 0, \]
where \(\sum _{r=1}^n F_r \delta x_r = \sum _{i = 1} ^m Q _i \delta q ^i = Q \cdot \delta q\) is the virtual work done by the force field \(Q\) with a virtual displacement \(\delta q \ .\)
An important case of a nonpotential force is friction. In the case of so-called Rayleigh dissipation, \(Q_i\) is a linear coordinate-dependent combination of the system velocities \(\dot{q}^i\ .\)
There are many topics one can cover here but we do not have space for. One topic
is relativistic Lagrangians. see the Wikipedia article on Lagrangians
[[1]]. Gauge invariance is another topic that
is covered elsewhere. Infinite-dimensional Lagrangians are covered e.g. in Goldstein (1950).
Systems described by infinite-dimensional Lagrangians include wave systems and flexible rods.
Examples
Some simple but important examples of Lagrangians are as follows:
The free particle of mass \(m\) in three dimensions:
\[ L=\frac{1}{2}m(\dot{x}^2+\dot{y}^2+\dot{z}^2)\ .\]
The harmonic oscillator in one dimension:
\[ L=\frac{1}{2}m(\dot{x}^2-{x}^2)\ .\]
The Kepler (two body) problem in two dimensions:
\[ L=\frac{1}{2} \frac{m_1 m_2}{m_1 \!+ \!m_2} (\dot{x}^2+\dot{y}^2) + \frac{ G m_1 m_2 }{\sqrt{x^2+y^2}}\ .\]
References
- Arnold, V. (1989). Mathematical Methods of Classical Mechanics, Springer-Verlag, New York.
- Bloch, Anthony. (2003). Nonholonomic Mechanics and Control, Springer-Verlag, New York.
- Gelfand, I.M. and Fomin, S. (1963). Calculus of Variations, Prentice-Hall (reprinted by Dover, 2000), New Jersey.
- Goldstein, H. (1950). Classical Mechanics, Prentice-Hall, New York.
- Lagrange, J.L (1788). Mechanique Analytique, Chez la Veuve Desaint, Paris.
- Marsden, J.E. and Ratiu, T.S. (FirstEdition 1994, Second Edition, 1999). Introduction to Mechanics and Symmetry, Springer-Verlag, New York.
- Newton, I. (1687). Philosophi Naturalis Principia Mathematica, Josephi and Streater, London.
- Pars, L. (1965). Treatise on Analytical Mechanics, Heineman, London.
- Whitakker, E.T. (1988). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Cambridge University Press, Cambridge.
