Anthony Bloch
Mathematics, University of Michigan, Ann Arbor, MI
In development
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)).
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, both measured
with respect to an inertial frame.
Generalized Coordinates
{\bfi 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) \( x_r = x_r(q^1,q^2,\dots, q^m), \quad r = 1, \dots, n, \) 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.
Then one can show (the infinitesimal form of the Lagrange--d'Alembert principle)
\( \sum_{i=1}^m \left\{\sum_{r=1}^n(m_r\ddot{x_r}-F_r) \frac{\partial x_r}{\partial q ^i}\right\}\delta q ^i \)
become
\( \sum_{i=1}^m \left\{\frac{d}{dt} \frac{\partial T}{\partial \dot{ q} ^i}-\frac{\partial T}{\partial q^i}-Q_i \right\}\delta q^i = 0, \)
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 <mat>L=T-V</math> 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 Marsden and Ratiu, 1999.
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 \(T\) 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
\( \delta \int_a^b L (q (t), \dot{ q} (t))\,dt = 0 , \)
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 proof uses the integration by parts and the boundary conditions.
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.
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.\footnote 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 \(Q \cdot \delta q = \sum _{i = 1} ^m Q _i \delta q ^i\) is the virtual work done by the force field \(Q\) with a virtual displacement \(\delta q \).
Hamel's Equations
One can generalize Hamel's equations by writing them in a noncoordinate frame. This leads to Hamel's equations as discussed for example in (Bloch, Marsden and Zenkov 2009).
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.
- 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. .
