Analytical mechanics (or theoretical mechanics), developed in the 18th century and onward, are mathematical physics' refinements of classical mechanics, originally Newtonian mechanics, often termed vectorial mechanics. To model motion, analytical mechanics uses two scalar properties of motion—its kinetic energy and its potential energy—not Newton's vectorial forces. (A scalar is represented by a quantity, as denotes intensity, whereas a vector is represented by quantity plus direction.)
Principally Lagrangian mechanics and Hamiltonian mechanics, both tightly intertwined, analytical mechanics efficiently extends the scope of classical mechanics to solve problems by employing the concept of a system's constraints and path integrals. Using these concepts, theoretical physicists—such as Schrödinger, Dirac, Heisenberg and Feynman—developed quantum mechanics and its elaboration, quantum field theory. Applications and extensions reach into Einstein's general relativity as well as chaos theory. A very general result from analytical mechanics is Noether's theorem, which fuels much of modern theoretical physics.
- Generalized coordinates and constraints
In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a body's position during its motion. In physical systems, however, some structure or other system usually constrains the body's motion from taking certain directions and pathways. So a full set of Cartesian coordinates is often unneeded, as the constraints determine the evolving relations among the coordinates, which relations can be modeled by equations corresponding to the constraints. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motion's geometry, reducing the number of coordinates to the minimum needed to model the motion. These are known as generalized coordinates, denoted qi (i = 1, 2, 3...).
Generalized coordinates incorporate constraints on the system. There is one generalized coordinate qi for each degree of freedom (for convenience labelled by an index i = 1, 2...N), i.e. each way the system can change its configuration; as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates. The number of curvilinear coordinates equals the dimension of the position space in question (usually 3 for 3d space), while the number of generalized coordinates is not necessarily equal to this dimension; constraints can reduce the number of degrees of freedom (hence the number of generalized coordinates required to define the configuration of the system), following the general rule:
- [dimension of position space (usually 3)] × [number of constituents of system ("particles")] − (number of constraints)
- = (number of degrees of freedom) = (number of generalized coordinates)
For a system with N degrees of freedom, the generalized coordinates can be collected into an N-tuple:
and the time derivative (here denoted by an overdot) of this tuple give the generalized velocities:
The foundation which the subject is built on is D'Alembert's principle.
This principle states that infinitesimal virtual work done by a force is zero, which is the work done by a force consistent with the constraints of the system. The idea of a constraint is useful - since this limits what the system can do, and can provide steps to solving for the motion of the system. The equation for D'Alembert's principle is:
are the generalized forces (script Q instead of ordinary Q is used here to prevent conflict with canonical transformations below) and q are the generalized coordinates. This leads to the generalized form of Newton's laws in the language of analytical mechanics:
where T is the total kinetic energy of the system, and the notation
is a useful shorthand (see matrix calculus for this notation).
If the curvilinear coordinate system is defined by the standard position vector r, and if the position vector can be written in terms of the generalized coordinates q and time t in the form:
and this relation holds for all times t, then q are called Holonomic constraints. Vector r is explicitly dependent on t in cases when the constraints vary with time, not just because of q(t). For time-independent situations, the constraints are also called scleronomic, for time-dependent cases they are called rheonomic.
The introduction of generalized coordinates and the fundamental Lagrangian function:
where T is the total kinetic energy and V is the total potential energy of the entire system, then either following the calculus of variations or using the above formula - lead to the Euler–Lagrange equations;
which are a set of N second-order ordinary differential equations, one for each qi(t).
This formulation identifies the actual path followed by the motion as a selection of the path over which the time integral of kinetic energy is least, assuming the total energy to be fixed, and imposing no conditions on the time of transit.
The Lagrangian formulation uses the configuration space of the system, the set of all possible generalized coordinates:
where is N-dimensional real space (see also set-builder notation). The particular solution to the Euler–Lagrange equations is called a (configuration) path or trajectory, i.e. one particular q(t) subject to the required initial conditions. The general solutions form a set of possible configurations as functions of time:
The Legendre transformation of the Lagrangian replaces the generalized coordinates and velocities (q, q̇) with (q, p); the generalized coordinates and the generalized momenta conjugate to the generalized coordinates:
and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):
which are now a set of 2N first-order ordinary differential equations, one for each qi(t) and pi(t). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:
which is often considered one of Hamilton's equations of motion additionally to the others. The generalized momenta can be written in terms of the generalized forces in the same way as Newton's second law:
Generalized momentum space
Analogous to the configuration space, the set of all momenta is the momentum space (technically in this context; generalized momentum space):
The set of all positions and momenta form the phase space;
that is, the cartesian product × of the configuration space and generalized momentum space.
A particular solution to Hamilton's equations is called a phase path, i.e. a particular curve (q(t),p(t)), subject to the required initial conditions. The set of all phase paths, i.e. general solution to the differential equations, is the phase portrait:
Likewise, the phase space can be defined more deeply using topological manifolds and the cotangent bundle.
The Hamiltonian formulation is more general, allowing time-varying energy, identifying the path followed to be the one with stationary action. This is known as the principle of least action:
holding the departure t1 and arrival t2 times fixed. The term action has various meanings. This definition is only one, and corresponds specifically to an integral of the Lagrangian of the system. The term "path" or "trajectory" refers to the time evolution of the system as a path through configuration space , i.e. q(t) tracing out a path in . The path for which action is least is the path taken by the system.
From this principle, all equations of motion in classical mechanics can be derived. Generalizations of these approaches underlie the path integral formulation of quantum mechanics, and is used for calculating geodesic motion in general relativity.
Properties of the Lagrangian and Hamiltonian functions
- All the individual generalized coordinates qi(t), velocities q̇i(t) and momenta pi(t) for every degree of freedom are mutually independent. Explicit time-dependence of a function means the function actually includes time t as a variable in addition to the q(t), p(t), not simply as a parameter through q(t) and p(t), which would mean explicit time-independence.
- The Lagrangian is invariant under addition of the total time derivative of any function of q and t, that is:
- so each Lagrangian L and L' describe exactly the same motion.
- Analogously, the Hamiltonian is invariant under addition of the partial time derivative of any function of q, p and t, that is:
- (K is a frequently used letter in this case). This property is used in canonical transformations (see below).
- If the Lagrangian is independent of some generalized coordinates, then the generalized momenta conjugate to those coordinates are constants of the motion, i.e. are conserved, this immediately follows from Lagrange's equations:
- Such coordinates are "cyclic" or "ignorable". It can be shown that the Hamiltonian is also cyclic in exactly the same generalized coordinates.
- If the Lagrangian is time-independent the Hamiltonian is also time-independent (i..e both are constant in time).
- If the kinetic energy is a homogeneous function (of degree 2 - quadratic) of the generalized velocities and the Lagrangian is explicitly time-independent:
- where λ is a constant, then the Hamiltonian will be the total conserved energy, equal to the total the kinetic and potential energies of the system:
The invariance of the Hamiltonian (under addition of the partial time derivative of an arbitrary function of p, q, and t) allows the Hamiltonian in one set of coordinates q and momenta p to be transformed into a new set Q = Q(q, p, t) and P = P(q, p, t), in four possible ways:
With the restriction on P and Q such that the transformed Hamiltonian system is:
the above transformations are called canonical transformations, each function Gn is called a generating function of the "nth kind" or "type-n". The transformation of coordinates and momenta can allow simplification for solving Hamilton's equations for a given problem.
- The Poisson bracket
The choice of Q and P is completely arbitrary, but not every choice leads to a canonical transformation. A simple test to check if a transformation q → Q and p → P is canonical is to calculate the Poisson bracket, defined by:
and if it is unity:
for all i = 1, 2,...N, then the transformation is canonical, else it is not.
All dynamical variables can be derived from position r, momentum p, and time t, and written as a function of these: A = A(q, p, t). Calculating the total derivative of A and substituting Hamilton's equations into the result leads to the time evolution of A:
This equation in A is closely related to the equation of motion in the Heisenberg picture of quantum mechanics, in which classical dynamical variables become quantum operators (indicated by hats (^)), and the Poission bracket is replaced by the commutator of operators via Dirac's canonical quantization:
By setting the canonically transformed Hamiltonian K = 0, and the type-2 generating function equal to Hamilton's principal function (also the action ) plus an arbitrary constant C:
the generalized momenta become:
and P is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:
where H is the Hamiltonian as before:
Another related function is Hamilton's characteristic function
used to solve the HJE by additive separation of variables for a time-independent Hamiltonian H.
The study of the solutions of the Hamilton-Jacobi equations leads naturally to the study of symplectic manifolds and symplectic topology. In this formulation, the solutions of the Hamilton–Jacobi equations are the integral curves of Hamiltonian vector fields.
Extensions to classical field theory
- Lagrangian field theory
where ∂μ denotes the 4-gradient, the Euler–Lagrange equations can be extended to fields:
Originally developed for classical fields, the above formulation is applicable to all physical fields in classical, quantum, and relativistic situations: such as Newtonian gravity, classical electromagnetism, general relativity, and quantum field theory. It is a question of determining the correct Lagrangian density to generate the correct field equation.
- Hamiltonian field theory
The corresponding momentum field density conjugate to the field φ(r, t) is:
The Hamiltonian density (Hamiltonian per unit volume) is likewise;
and satisfies analogously:
which is like a Lagrangian, only with N − 1 degrees of freedom. The Routhian density satisfies:
Symmetry, conservation, and Noether's theorem
- Symmetry transformations in classical space and time
Each transformation can be described by an operator (i.e. function acting on the position r or momentum p variables to change them). The following are the cases when the operator does not change r or p, i.e. symmetries.
Transformation Operator Position Momentum Translational symmetry Time translations Rotational invariance Galilean transformations Parity T-symmetry
Noether's theorem states that a continuous symmetry transformation of the action corresponds to a conservation law, i.e. the action (and hence the Lagrangian) doesn't change under a transformation parameterized by a parameter s:
the Lagrangian describes the same motion independent of s, which can be length, angle of rotation, or time. The corresponding momenta to q will be conserved.
References and notes
- Lanczos, Cornelius (1970). The variational principles of mechanics (4th ed.). New York: Dover Publications Inc. Introduction, pp. xxi–xxix. ISBN 0-486-65067-7.
- The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0-679-77631-1
- Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 978-0-521-57572-0
- McGraw Hill Encyclopaedia of Physics (2nd Edition), C.B. Parker, 1994, ISBN 0-07-051400-3
- Penrose, R. (2007). The Road to Reality. Vintage books. p. 474. ISBN 0-679-77631-1.
- Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
- Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
- Quantum Field Theory, D. McMahon, Mc Graw Hill (US), 2008, ISBN 978-0-07-154382-8
- Relativity, Gravitation, and Cosmology, R.J.A. Lambourne, Open University, Cambridge University Press, 2010, ISBN 978-0-521-13138-4
- Classical Mechanics, T.W.B. Kibble, European Physics Series, McGraw-Hill (UK), 1973, ISBN 0-07-084018-0
- Arnolʹd, VI (1989). Mathematical methods of classical mechanics (2nd Edition ed.). Springer. Chapter 8. ISBN 978-0-387-96890-2.
- Doran, C; Lasenby, A (2003). Geometric algebra for physicists. Cambridge University Press. p. §12.3, pp. 432–439. ISBN 978-0-521-71595-9.
- Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, ISBN 0-7167-0344-0
- Action (physics)
- Applied mechanics
- Classical mechanics
- Hamilton–Jacobi equation
- Hamilton's principle
- Kinetics (physics)
- Non-autonomous mechanics