Analytical mechanics

Still, though lacking precise definitions, it is obvious that the two-body problem has a simple solution, whereas the three-body problem has not. The two-body problem is solved by formulas involving parameters; their values can be changed to study the class of all solutions, that is, the mathematical structure of the problem. Moreover, an accurate mental or drawn picture can be made for the motion of two bodies, and it can be as real and accurate as the real bodies moving and interacting. In the three-body problem, parameters can also be assigned specific values; however, the solution at these assigned values or a collection of such solutions does not reveal the mathematical structure of the problem. As in many other problems, the mathematical structure can be elucidated only by examining the differential equations themselves.

Analytical mechanics aims at even more: not at understanding the mathematical structure of a single mechanical problem, but that of a class of problems so wide that they encompass most of mechanics. It concentrates on systems to which Lagrangian or Hamiltonian equations of motion are applicable and that include a very wide range of problems indeed.^[3]

Development of analytical mechanics has two objectives: (i) increase the range of solvable problems by developing standard techniques with a wide range of applicability, and (ii) understand the mathematical structure of mechanics. In the long run, however, (ii) can help (i) more than a concentration on specific problems for which methods have already been designed.

In

Generalized coordinates incorporate constraints on the system. There is one generalized coordinate q_i 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:^{[5][dubious – discuss]}

For a system with N degrees of freedom, the generalized coordinates can be collected into an N-tuple: $${\displaystyle \mathbf {q} =(q_{1},q_{2},\dots ,q_{N})}$$ and the time derivative (here denoted by an overdot) of this tuple give the generalized velocities: $${\displaystyle {\frac {d\mathbf {q} }{dt}}=\left({\frac {dq_{1}}{dt}},{\frac {dq_{2}}{dt}},\dots ,{\frac {dq_{N}}{dt}}\right)\equiv \mathbf {\dot {q}} =({\dot {q}}_{1},{\dot {q}}_{2},\dots ,{\dot {q}}_{N}).}$$

D'Alembert's principle states that infinitesimal virtual work done by a force across reversible displacements is zero, which is the work done by a force consistent with ideal 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:^[6]: 265 $${\displaystyle \delta W={\boldsymbol {\mathcal {Q}}}\cdot \delta \mathbf {q} =0\,,}$$ where $${\displaystyle {\boldsymbol {\mathcal {Q}}}=({\mathcal {Q}}_{1},{\mathcal {Q}}_{2},\dots ,{\mathcal {Q}}_{N})}$$ are the

$${\displaystyle {\boldsymbol {\mathcal {Q}}}={\frac {d}{dt}}\left({\frac {\partial T}{\partial \mathbf {\dot {q}} }}\right)-{\frac {\partial T}{\partial \mathbf {q} }}\,,}$$

where T is the total kinetic energy of the system, and the notation $${\displaystyle {\frac {\partial }{\partial \mathbf {q} }}=\left({\frac {\partial }{\partial q_{1}}},{\frac {\partial }{\partial q_{2}}},\dots ,{\frac {\partial }{\partial q_{N}}}\right)}$$ is a useful shorthand (see matrix calculus for this notation).

If the curvilinear coordinate system is defined by the standard

$${\displaystyle \mathbf {r} =\mathbf {r} (\mathbf {q} (t),t)}$$

The introduction of generalized coordinates and the fundamental Lagrangian function:

${\displaystyle L(\mathbf {q} ,\mathbf {\dot {q}} ,t)=T(\mathbf {q} ,\mathbf {\dot {q}} ,t)-V(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

where T is the total

${\displaystyle {\frac {d}{dt}}\left({\frac {\partial L}{\partial \mathbf {\dot {q}} }}\right)={\frac {\partial L}{\partial \mathbf {q} }}\,,}$

which are a set of N second-order ordinary differential equations, one for each q_i(t).

This formulation identifies the actual path followed by the motion as a selection of the path over which the

The Lagrangian formulation uses the configuration space of the system, the set of all possible generalized coordinates:

${\displaystyle {\mathcal {C}}=\{\mathbf {q} \in \mathbb {R} ^{N}\}\,,}$

where ${\displaystyle \mathbb {R} ^{N}}$ is N-dimensional

${\displaystyle \{\mathbf {q} (t)\in \mathbb {R} ^{N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {C}}\,,}$

The configuration space can be defined more generally, and indeed more deeply, in terms of topological manifolds and the tangent bundle.

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:

${\displaystyle \mathbf {p} ={\frac {\partial L}{\partial \mathbf {\dot {q}} }}=\left({\frac {\partial L}{\partial {\dot {q}}_{1}}},{\frac {\partial L}{\partial {\dot {q}}_{2}}},\cdots {\frac {\partial L}{\partial {\dot {q}}_{N}}}\right)=(p_{1},p_{2}\cdots p_{N})\,,}$

and introduces the Hamiltonian (which is in terms of generalized coordinates and momenta):

${\displaystyle H(\mathbf {q} ,\mathbf {p} ,t)=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {\dot {q}} ,t)}$

where ${\displaystyle \cdot }$ denotes the dot product, also leading to Hamilton's equations:

${\displaystyle \mathbf {\dot {p}} =-{\frac {\partial H}{\partial \mathbf {q} }}\,,\quad \mathbf {\dot {q}} =+{\frac {\partial H}{\partial \mathbf {p} }}\,,}$

which are now a set of 2N first-order ordinary differential equations, one for each q_i(t) and p_i(t). Another result from the Legendre transformation relates the time derivatives of the Lagrangian and Hamiltonian:

${\displaystyle {\frac {dH}{dt}}=-{\frac {\partial L}{\partial t}}\,,}$

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:

${\displaystyle \mathbf {\dot {p}} ={\boldsymbol {\mathcal {Q}}}\,.}$

Analogous to the configuration space, the set of all momenta is the generalized

${\displaystyle {\mathcal {M}}=\{\mathbf {p} \in \mathbb {R} ^{N}\}\,.}$

("Momentum space" also refers to "k-space"; the set of all

The set of all positions and momenta form the phase space:

${\displaystyle {\mathcal {P}}={\mathcal {C}}\times {\mathcal {M}}=\{(\mathbf {q} ,\mathbf {p} )\in \mathbb {R} ^{2N}\}\,,}$

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, a particular curve (q(t),p(t)) subject to the required initial conditions. The set of all phase paths, the general solution to the differential equations, is the phase portrait:

${\displaystyle \{(\mathbf {q} (t),\mathbf {p} (t))\in \mathbb {R} ^{2N}\,:\,t\geq 0,t\in \mathbb {R} \}\subseteq {\mathcal {P}}\,,}$

All dynamical variables can be derived from position q, momentum p, and time t, and written as a function of these: A = A(q, p, t). If A(q, p, t) and B(q, p, t) are two scalar valued dynamical variables, the Poisson bracket is defined by the generalized coordinates and momenta:

${\displaystyle {\begin{aligned}\{A,B\}\equiv \{A,B\}_{\mathbf {q} ,\mathbf {p} }&={\frac {\partial A}{\partial \mathbf {q} }}\cdot {\frac {\partial B}{\partial \mathbf {p} }}-{\frac {\partial A}{\partial \mathbf {p} }}\cdot {\frac {\partial B}{\partial \mathbf {q} }}\\&\equiv \sum _{k}{\frac {\partial A}{\partial q_{k}}}{\frac {\partial B}{\partial p_{k}}}-{\frac {\partial A}{\partial p_{k}}}{\frac {\partial B}{\partial q_{k}}}\,,\end{aligned}}}$

Calculating the total derivative of one of these, say A, and substituting Hamilton's equations into the result leads to the time evolution of A:

${\displaystyle {\frac {dA}{dt}}=\{A,H\}+{\frac {\partial A}{\partial t}}\,.}$

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 Poisson bracket is replaced by the commutator of operators via Dirac's canonical quantization:

${\displaystyle \{A,B\}\rightarrow {\frac {1}{i\hbar }}[{\hat {A}},{\hat {B}}]\,.}$

Following are overlapping properties between the Lagrangian and Hamiltonian functions.^[5][8]

• All the individual generalized coordinates q_i(t), velocities q̇_i(t) and momenta p_i(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: $${\displaystyle L'=L+{\frac {d}{dt}}F(\mathbf {q} ,t)\,,}$$ so each Lagrangian L and L describe exactly the same motion. In other words, the Lagrangian of a system is not unique.
• Analogously, the Hamiltonian is invariant under addition of the partial time derivative of any function of q, p and t, that is: $${\displaystyle K=H+{\frac {\partial }{\partial t}}G(\mathbf {q} ,\mathbf {p} ,t)\,,}$$ (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: $${\displaystyle {\frac {\partial L}{\partial q_{j}}}=0\,\rightarrow \,{\frac {dp_{j}}{dt}}={\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{j}}}=0}$$ 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 of the generalized velocities, and the Lagrangian is explicitly time-independent, then: $${\displaystyle T((\lambda {\dot {q}}_{i})^{2},(\lambda {\dot {q}}_{j}\lambda {\dot {q}}_{k}),\mathbf {q} )=\lambda ^{2}T(({\dot {q}}_{i})^{2},{\dot {q}}_{j}{\dot {q}}_{k},\mathbf {q} )\,,\quad L(\mathbf {q} ,\mathbf {\dot {q}} )\,,}$$ where λ is a constant, then the Hamiltonian will be the total conserved energy, equal to the total kinetic and potential energies of the system: $${\displaystyle H=T+V=E\,.}$$ This is the basis for the
  quantum operators
  directly obtains it.

Action is another quantity in analytical mechanics defined as a functional of the Lagrangian:

${\displaystyle {\mathcal {S}}=\int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt\,.}$

A general way to find the equations of motion from the action is the

where the departure t₁ and arrival t₂ times are fixed.[1] The term "path" or "trajectory" refers to the time evolution of the system as a path through configuration space ${\displaystyle {\mathcal {C}}}$, in other words q(t) tracing out a path in ${\displaystyle {\mathcal {C}}}$. 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. This approach can be extended to fields rather than a system of particles (see below), and underlies the path integral formulation of quantum mechanics,^[11][12] and is used for calculating geodesic motion in general relativity.^[13]

Canonical transformations

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:

${\displaystyle {\begin{aligned}&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{1}(\mathbf {q} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{2}(\mathbf {q} ,\mathbf {P} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{3}(\mathbf {p} ,\mathbf {Q} ,t)\\&K(\mathbf {Q} ,\mathbf {P} ,t)=H(\mathbf {q} ,\mathbf {p} ,t)+{\frac {\partial }{\partial t}}G_{4}(\mathbf {p} ,\mathbf {P} ,t)\\\end{aligned}}}$

With the restriction on P and Q such that the transformed Hamiltonian system is:

${\displaystyle \mathbf {\dot {P}} =-{\frac {\partial K}{\partial \mathbf {Q} }}\,,\quad \mathbf {\dot {Q}} =+{\frac {\partial K}{\partial \mathbf {P} }}\,,}$

the above transformations are called canonical transformations, each function G_n 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 choice of Q and P is completely arbitrary, but not every choice leads to a canonical transformation. One simple criterion for a transformation q → Q and p → P to be canonical is the Poisson bracket be unity,

${\displaystyle \{Q_{i},P_{i}\}=1}$

for all i = 1, 2,...N. If this does not hold then the transformation is not canonical.^[5]

The Hamilton–Jacobi equation

By setting the canonically transformed Hamiltonian K = 0, and the type-2 generating function equal to Hamilton's principal function (also the action ${\displaystyle {\mathcal {S}}}$) plus an arbitrary constant C:

${\displaystyle G_{2}(\mathbf {q} ,t)={\mathcal {S}}(\mathbf {q} ,t)+C\,,}$

the generalized momenta become:

${\displaystyle \mathbf {p} ={\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }}}$

and P is constant, then the Hamiltonian-Jacobi equation (HJE) can be derived from the type-2 canonical transformation:

${\displaystyle H=-{\frac {\partial {\mathcal {S}}}{\partial t}}}$

where H is the Hamiltonian as before:

${\displaystyle H=H(\mathbf {q} ,\mathbf {p} ,t)=H\left(\mathbf {q} ,{\frac {\partial {\mathcal {S}}}{\partial \mathbf {q} }},t\right)}$

Another related function is Hamilton's characteristic function

${\displaystyle W(\mathbf {q} )={\mathcal {S}}(\mathbf {q} ,t)+Et}$

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

Routhian mechanics is a hybrid formulation of Lagrangian and Hamiltonian mechanics, not often used but especially useful for removing cyclic coordinates.^{[citation needed]} If the Lagrangian of a system has s cyclic coordinates q = q₁, q₂, ... q_s with conjugate momenta p = p₁, p₂, ... p_s, with the rest of the coordinates non-cyclic and denoted ζ = ζ₁, ζ₁, ..., ζ_{N − s}, they can be removed by introducing the Routhian:

${\displaystyle R=\mathbf {p} \cdot \mathbf {\dot {q}} -L(\mathbf {q} ,\mathbf {p} ,{\boldsymbol {\zeta }},{\dot {\boldsymbol {\zeta }}})\,,}$

which leads to a set of 2s Hamiltonian equations for the cyclic coordinates q,

${\displaystyle {\dot {\mathbf {q} }}=+{\frac {\partial R}{\partial \mathbf {p} }}\,,\quad {\dot {\mathbf {p} }}=-{\frac {\partial R}{\partial \mathbf {q} }}\,,}$

and N − s Lagrangian equations in the non cyclic coordinates ζ.

${\displaystyle {\frac {d}{dt}}{\frac {\partial R}{\partial {\dot {\boldsymbol {\zeta }}}}}={\frac {\partial R}{\partial {\boldsymbol {\zeta }}}}\,.}$

Set up in this way, although the Routhian has the form of the Hamiltonian, it can be thought of a Lagrangian with N − s degrees of freedom.

The coordinates q do not have to be cyclic, the partition between which coordinates enter the Hamiltonian equations and those which enter the Lagrangian equations is arbitrary. It is simply convenient to let the Hamiltonian equations remove the cyclic coordinates, leaving the non cyclic coordinates to the Lagrangian equations of motion.

Appell's equation of motion involve generalized accelerations, the second time derivatives of the generalized coordinates:

${\displaystyle \alpha _{r}={\ddot {q}}_{r}={\frac {d^{2}q_{r}}{dt^{2}}}\,,}$

as well as generalized forces mentioned above in D'Alembert's principle. The equations are

${\displaystyle {\mathcal {Q}}_{r}={\frac {\partial S}{\partial \alpha _{r}}}\,,\quad S={\frac {1}{2}}\sum _{k=1}^{N}m_{k}\mathbf {a} _{k}^{2}\,,}$

where

${\displaystyle \mathbf {a} _{k}={\ddot {\mathbf {r} }}_{k}={\frac {d^{2}\mathbf {r} _{k}}{dt^{2}}}}$

is the acceleration of the k particle, the second time derivative of its position vector. Each acceleration a_k is expressed in terms of the generalized accelerations α_r, likewise each r_k are expressed in terms the generalized coordinates q_r.

Generalized coordinates apply to discrete particles. For N

$${\displaystyle {\mathcal {L}}={\mathcal {L}}(\phi _{1},\phi _{2},\dots ,\nabla \phi _{1},\nabla \phi _{2},\dots ,\partial _{t}\phi _{1},\partial _{t}\phi _{2},\ldots ,\mathbf {r} ,t)\,.}$$

$${\displaystyle \partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi _{i})}}\right)={\frac {\partial {\mathcal {L}}}{\partial \phi _{i}}}\,,}$$

This scalar field formulation can be extended to

The Lagrangian is the volume integral of the Lagrangian density:^[12][16] $${\displaystyle L=\int _{\mathcal {V}}{\mathcal {L}}\,dV\,.}$$

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.

The corresponding "momentum" field densities conjugate to the N scalar fields φ_i(r, t) are:^[12] $${\displaystyle \pi _{i}(\mathbf {r} ,t)={\frac {\partial {\mathcal {L}}}{\partial {\dot {\phi }}_{i}}}\,\quad {\dot {\phi }}_{i}\equiv {\frac {\partial \phi _{i}}{\partial t}}}$$ where in this context the overdot denotes a partial time derivative, not a total time derivative. The Hamiltonian density ${\displaystyle {\mathcal {H}}}$ is defined by analogy with mechanics: $${\displaystyle {\mathcal {H}}(\phi _{1},\phi _{2},\ldots ,\pi _{1},\pi _{2},\ldots ,\mathbf {r} ,t)=\sum _{i=1}^{N}{\dot {\phi }}_{i}(\mathbf {r} ,t)\pi _{i}(\mathbf {r} ,t)-{\mathcal {L}}\,.}$$

The equations of motion are: $${\displaystyle {\dot {\phi }}_{i}=+{\frac {\delta {\mathcal {H}}}{\delta \pi _{i}}}\,,\quad {\dot {\pi }}_{i}=-{\frac {\delta {\mathcal {H}}}{\delta \phi _{i}}}\,,}$$ where the

$${\displaystyle {\frac {\delta }{\delta \phi _{i}}}={\frac {\partial }{\partial \phi _{i}}}-\partial _{\mu }{\frac {\partial }{\partial (\partial _{\mu }\phi _{i})}}}$$

Again, the volume integral of the Hamiltonian density is the Hamiltonian $${\displaystyle H=\int _{\mathcal {V}}{\mathcal {H}}\,dV\,.}$$

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.^[11]

Transformation	Operator	Position	Momentum
Translational symmetry	${\displaystyle X(\mathbf {a} )}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {a} }$	${\displaystyle \mathbf {p} \rightarrow \mathbf {p} }$
Time translation	${\displaystyle U(t_{0})}$	${\displaystyle \mathbf {r} (t)\rightarrow \mathbf {r} (t+t_{0})}$	${\displaystyle \mathbf {p} (t)\rightarrow \mathbf {p} (t+t_{0})}$
Rotational invariance	${\displaystyle R(\mathbf {\hat {n}} ,\theta )}$	${\displaystyle \mathbf {r} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {r} }$	${\displaystyle \mathbf {p} \rightarrow R(\mathbf {\hat {n}} ,\theta )\mathbf {p} }$
Galilean transformations	${\displaystyle G(\mathbf {v} )}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} +\mathbf {v} t}$	${\displaystyle \mathbf {p} \rightarrow \mathbf {p} +m\mathbf {v} }$
Parity	${\displaystyle P}$	${\displaystyle \mathbf {r} \rightarrow -\mathbf {r} }$	${\displaystyle \mathbf {p} \rightarrow -\mathbf {p} }$
T-symmetry	${\displaystyle T}$	${\displaystyle \mathbf {r} \rightarrow \mathbf {r} (-t)}$	${\displaystyle \mathbf {p} \rightarrow -\mathbf {p} (-t)}$

where R(n̂, θ) is the rotation matrix about an axis defined by the unit vector n̂ and angle θ.

Noether's theorem

Noether's theorem states that a

s: $${\displaystyle L[q(s,t),{\dot {q}}(s,t)]=L[q(t),{\dot {q}}(t)]}$$ 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.

• Lagrangian mechanics
• Hamiltonian mechanics
• Theoretical mechanics
• Classical mechanics
• Hamilton–Jacobi equation
• Hamilton's principle
• Kinematics
• Kinetics (physics)
• Non-autonomous mechanics
• Udwadia–Kalaba equation^{[neutrality is disputed}
  ]

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