</td></tr></table>
Lagrangian mechanics is a reformulation of classical mechanics using the principle of stationary action (also called the principle of least action).^{[1]} Lagrangian mechanics applies to systems whether or not they conserve energy or momentum, and it provides conditions under which energy, momentum or both are conserved.^{[2]} It was introduced by the ItalianFrench mathematician JosephLouis Lagrange in 1788.
In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind,^{[3]} which treat constraints explicitly as extra equations, often using Lagrange multipliers;^{[4]}^{[5]} or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.^{[3]}^{[6]} The fundamental lemma of the calculus of variations shows that solving the Lagrange equations is equivalent to finding the path for which the action functional is stationary, a quantity that is the integral of the Lagrangian over time.
The use of generalized coordinates may considerably simplify a system's analysis. For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics would require solving for the timevarying constraint force required to keep the bead in the groove. For the same problem using Lagrangian mechanics, one looks at the path of the groove and chooses a set of independent generalized coordinates that completely characterize the possible motion of the bead. This choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the groove on the bead at a given moment.
Lagrangian mechanics ability to work in generalized coordinate systems makes it suitable for use in special relativity which coordinate systems change for different observers. It is used as the standard language of particle physics to express the standard model in terms of Lagrangians.
Conceptual framework
Generalized coordinates
Concepts and terminology
For one particle acted on by external forces, Newton's second law forms a set of 3 secondorder ordinary differential equations, one for each dimension. Therefore, the motion of the particle can be completely described by 6 independent variables: 3 initial position coordinates and 3 initial velocity coordinates. Given these, the general solutions to Newton's second law become particular solutions that determine the time evolution of the particle's behaviour after its initial state (t = 0).
The most familiar set of variables for position r = (r_{1}, r_{2}, r_{3}) and velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{\dot{r}}_j = (\dot{r}_1, \dot{r}_2, \dot{r}_3)}
are Cartesian coordinates and their time derivatives (i.e. position (x, y, z) and velocity (v_{x}, v_{y}, v_{z}) components). Determining forces in terms of standard coordinates can be complicated, and usually requires much labour.
An alternative and more efficient approach is to use only as many coordinates as are needed to define the position of the particle, at the same time incorporating the constraints on the system, and writing down kinetic and potential energies. In other words, to determine the number of degrees of freedom the particle has, i.e. the number of possible ways the system can move subject to the constraints (forces that prevent it moving in certain paths). Energies are much easier to write down and calculate than forces, since energy is a scalar while forces are vectors.
These coordinates are generalized coordinates, denoted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle q_j}
, and there is one for each degree of freedom. Their corresponding time derivatives are the generalized velocities, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{q_j}}
. The number of degrees of freedom is usually not equal to the number of spatial dimensions: multibody systems in 3dimensional space (such as Barton's Pendulums, planets in the solar system, or atoms in molecules) can have many more degrees of freedom incorporating rotations as well as translations. This contrasts the number of spatial coordinates used with Newton's laws above.
Mathematical formulation
The position vector r in a standard coordinate system (like Cartesian, spherical etc.), is related to the generalized coordinates by some transformation equation:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{r} = \bold{r}(q_i, t). \, }
where there are as many q_{i} as needed (number of degrees of freedom in the system). Likewise for velocity and generalized velocities.
For example, for a simple pendulum of length ℓ, there is the constraint of the pendulum bob's suspension (rod/wire/string etc.). The position r depends on the x and y coordinates at time t, that is, r(t)=(x(t),y(t)), however x and y are coupled to each other in a constraint equation (if x changes y must change, and vice versa). A logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, so we have r = (x(θ), y(θ)) = r(θ), in which θ = θ(t). Then the transformation equation would be
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{r}(\theta(t)) =(\ell\sin\theta, \ell\cos\theta)}
and so
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{\dot{r}}(\theta(t),\dot{\theta}(t))=( \ell\, \dot{\theta}\cos\theta, \ell\,\dot{\theta}\sin \theta)}
which corresponds to the one degree of freedom the pendulum has. The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system.
In general, from m independent generalized coordinates q_{j}, the following transformation equations hold for a system composed of n particles:^{[7]}^{:260}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \begin{array}{r c l} \mathbf{r}_1 &=& \mathbf{r}_1(q_1, q_2, \cdots, q_m, t) \\ \mathbf{r}_2 &=& \mathbf{r}_2(q_1, q_2, \cdots, q_m, t) \\ & \vdots & \\ \mathbf{r}_n &=& \mathbf{r}_n(q_1, q_2, \cdots, q_m, t) \end{array}}
where m indicates the total number of generalized coordinates. An expression for the virtual displacement (infinitesimal), δr_{i} of the system for timeindependent constraints or "velocitydependent constraints" is the same form as a total differential^{[7]}^{:264}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j,}
where j is an integer label corresponding to a generalized coordinate.
The generalized coordinates form a discrete set of variables that define the configuration of a system. The continuum analogue for defining a field are field variables, say ϕ(r, t), which represents density function varying with position and time.
D'Alembert's principle and generalized forces
D'Alembert's principle introduces the concept of virtual work due to applied forces F_{i} and inertial forces, acting on a threedimensional accelerating system of n particles whose motion is consistent with its constraints,^{[7]}^{:269}
Mathematically, the virtual work δW done on a particle of mass m_{i} through a virtual displacement δr_{i} (consistent with the constraints) is:
D'Alembert's principle
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta W = \sum_{i=1}^n ( \mathbf {F}_{i}  m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0.}

where a_{i} are the accelerations of the particles in the system and i = 1, 2,...,n simply labels the particles. In terms of generalized coordinates
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta W = \sum_{j=1}^m \sum_{i=1}^n ( \mathbf {F}_{i}  m_i \mathbf{a}_i )\cdot \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j= 0.}
this expression suggests that the applied forces may be expressed as generalized forces, Q_{j}. Dividing by δq_{j} gives the definition of a generalized force:^{[7]}^{:265}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q_j = \frac{\delta W}{\delta q_j}= \sum_{i=1}^n \mathbf {F}_i \cdot \frac {\partial \mathbf{r}_i} {\partial q_j}.}
If the forces F_{i} are conservative, there is a scalar potential field V in which the gradient of V is the force:^{[7]}^{:266 & 270}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf F_i =  \nabla V \Rightarrow Q_j =  \sum_{i=1}^n \nabla V \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} =  \frac {\partial V}{\partial q_j}.}
i.e. generalized forces can be reduced to a potential gradient in terms of generalized coordinates. The previous result may be easier to see by recognizing that V is a function of the r_{i}, which are in turn functions of q_{j}, and then applying the chain rule to the derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V}
with respect to q_{j}.
Kinetic energy relations
The kinetic energy, T, for the system of particles is defined by^{[7]}^{:269}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle T = \frac {1}{2} \sum_{i=1}^n m_i \mathbf {\dot{r}}_i \cdot \mathbf {\dot{r}}_i.}
The partial derivatives of T with respect to the generalized coordinates q_{j} and generalized velocities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{q}_j}
are:^{[7]}^{:269}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\partial T}{\partial q_j} = \sum_{i=1}^n m_i \mathbf{\dot{r}}_i \cdot \frac{\partial \mathbf{\dot{r}}_i}{\partial q_j}}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \quad \frac{\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf{\dot{r}}_i \cdot \frac{\partial \mathbf{\dot{r}}_i}{\partial \dot{q}_j}.}
Because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{q_j}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle q_j}
are independent variables:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac {\partial \mathbf{\dot{r}}_i}{\partial \dot{q_j}} = \frac {\partial \mathbf{r}_i}{\partial q_j} .}
Then:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \quad \frac{\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf{\dot{r}}_i \cdot \frac{\partial \mathbf{r}_i}{\partial q_j} \ .}
The total time derivative of this equation is
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf{\ddot{r}}_i \cdot \frac {\partial \mathbf{r}_i}{\partial q_j} + \sum_{i=1}^n m_i \mathbf{\dot{r}}_i \cdot \frac {\partial \mathbf{\dot{r}}_i}{\partial q_j} = Q_j + \frac{\partial T}{\partial q_j} \ .}
resulting in:
Generalized equations of motion
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q_j = \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right )  \frac {\partial T}{\partial q_j}}

Newton's laws are contained in it, yet there is no need to find the constraint forces because virtual work and generalized coordinates (which account for constraints) are used. This equation in itself is not actually used in practice, but is a step towards deriving Lagrange's equations (see below).^{[8]}
Lagrangian and action
The core element of Lagrangian mechanics is the Lagrangian function, which summarizes the dynamics of the entire system in a very simple expression. The physics of analyzing a system is reduced to choosing the most convenient set of generalized coordinates, determining the kinetic and potential energies of the constituents of the system, then writing down the equation for the Lagrangian to use in Lagrange's equations. It is defined by ^{[9]}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L = T  V \,}
where T is the total kinetic energy and V is the total potential energy of the system.
The next fundamental element is the action Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{S}}
, defined as the time integral of the Lagrangian:^{[8]}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{S} = \int_{t_1}^{t_2} L\,\mathrm{d}t.}
This also contains the dynamics of the system, and has deep theoretical implications (discussed below). Technically, the action is a functional, that is, it is a function that maps the full Lagrangian function for all times between t_{1} and t_{2} to a scalar value for the action. Its dimensions are the same as angular momentum.
In classical field theory, the physical system is not a set of discrete particles, but rather a continuous field defined over a region of 3d space. Associated with the field is a Lagrangian density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{L}(\mathbf{r},t)}
defined in terms of the field and its derivatives at a location Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{r}}
. The total Lagrangian is then the integral of the Lagrangian density over 3d space (see volume integral):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L(t) = \int \mathcal{L}(\mathbf{r},t) \mathrm{d}^3 \mathbf{r} \,}
where d^{3}r is a 3d differential volume element, must be used instead. The action becomes an integral over space and time:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{S} = \int_{t_1}^{t_2}\int \mathcal{L}(\mathbf{r},t) \mathrm{d}^3\mathbf{r} \mathrm{d}t.}
Hamilton's principle of stationary action
Let q_{0} and q_{1} be the coordinates at respective initial and final times t_{0} and t_{1}. Using the calculus of variations, it can be shown that Lagrange's equations are equivalent to Hamilton's principle:
 The trajectory of the system between t_{0} and t_{1} has a stationary action S.
By stationary, we mean that the action does not vary to firstorder from infinitesimal deformations of the trajectory, with the endpoints (q_{0}, t_{0}) and (q_{1},t_{1}) fixed. Hamilton's principle can be written as:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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 the action functional need only be stationary, not necessarily a maximum or a minimum value. Any variation of the functional gives an increase in the functional integral of 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.
Lagrange equations of the first kind
Lagrange introduced an analytical method for finding stationary points using the method of Lagrange multipliers, and also applied it to mechanics.
For a system subject to the (holonomic) constraint equation on the generalized coordinates:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle F(r_1,r_2,r_3) = A }
where A is a constant, then Lagrange's equations of the first kind are:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \left[\frac{\partial L}{\partial r_j}  \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{r}_j}\right)\right] + \lambda\frac{\partial F}{\partial r_j}=0 }
where λ is the Lagrange multiplier. By analogy with the mathematical procedure, we can write:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\delta L}{\delta r_j} + \lambda\frac{\partial F}{\partial r_j}=0 }
where
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\delta L}{\delta r_j} = \frac{\partial L}{\partial r_j}  \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial \dot{r}_j}\right) }
denotes the variational derivative.
For e constraint equations F_{1}, F_{2},..., F_{e}, there is a Lagrange multiplier for each constraint equation, and Lagrange's equations of the first kind generalize to:
Lagrange's equations (1st kind)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\delta L}{\delta r_j} + \sum_{i=1}^e \lambda_i\frac{\partial F_i}{\partial r_j}=0 }

This procedure does increase the number of equations, but there are enough to solve for all of the multipliers. The number of equations generated is the number of constraint equations plus the number of coordinates, i.e. e + m. The advantage of the method is that (potentially complicated) substitution and elimination of variables linked by constraint equations can be bypassed.
There is a connection between the constraint equations F_{j} and the constraint forces N_{j} acting in the conservative system (forces are conservative):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle N_j = \sum_{i=1}^e \lambda_i \frac{\partial F_i}{\partial r_j} }
which is derived below.
Derivation of connection between constraint equations and forces

The generalized constraint forces are given by (using the definition of generalized force above):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle N_j = \sum_{i=1}^n \mathbf{N}_i\cdot\frac{\partial \mathbf{r}_i}{\partial q_j}}
and using the kinetic energy equation of motion (blue box above):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q_j = \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial T}{\partial \dot{q}_j} \right )  \frac {\partial T}{\partial q_j} = \frac{\delta T}{\delta q_j}=\sum_{i=1}^n \mathbf{F}_i\cdot\frac{\partial \mathbf{r}_i}{\partial q_j},}
For conservative systems (see below)
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathbf{F}_i = \nabla V_i + \mathbf{N}_i,}
so
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\delta T}{\delta q_j} = \sum_{i=1}^n \mathbf{F}_i\cdot\frac{\partial \mathbf{r}_i}{\partial q_j} =\sum_{i=1}^n (\nabla V_i + \mathbf{N}_i)\cdot\frac{\partial \mathbf{r}_i}{\partial q_j} =\sum_{i=1}^n\nabla V_i\cdot\frac{\partial \mathbf{r}_i}{\partial q_j}+\sum_{i=1}^n \mathbf{N}_i\cdot\frac{\partial \mathbf{r}_i}{\partial q_j} =\frac{\partial V}{\partial q_j} + N_j }
and
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\delta T}{\delta q_j}=\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac{\partial (L+V)}{\partial \dot{q}_j} \right )  \frac {\partial (L+V)}{\partial q_j} =\frac{\delta L}{\delta \dot{q}_j}  \frac {\partial V}{\partial q_j} }
equating leads to
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle N_j = \frac{\delta L}{\delta \dot{q}_j} }
and finally equating to Lagrange's equations of the first kind implies:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle N_j = \sum_{i=1}^e \lambda_i \frac{\partial F_i}{\partial r_j} }
So each constraint equation corresponds to a constraint force (in a conservative system).

Lagrange equations of the second kind
Euler–Lagrange equations
For any system with m degrees of freedom, the Lagrange equations include m generalized coordinates and m generalized velocities. Below, we sketch out the derivation of the Lagrange equations of the second kind. 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.
The equations of motion in Lagrangian mechanics are the Lagrange equations of the second kind, also known as the Euler–Lagrange equations:^{[8]}^{[10]}
Lagrange's equations (2nd kind)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} }

where j = 1, 2,...m represents the jth degree of freedom, q_{j} are the generalized coordinates, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{q}_j}
are the generalized velocities.
Although the mathematics required for Lagrange's equations appears significantly more complicated than Newton's laws, this does point to deeper insights into classical mechanics than Newton's laws alone: in particular, symmetry and conservation. In practice it's often easier to solve a problem using the Lagrange equations than Newton's laws, because the minimum generalized coordinates q_{i} can be chosen by convenience to exploit symmetries in the system, and constraint forces are incorporated into the geometry of the problem. There is one Lagrange equation for each generalized coordinate q_{i}.
For a system of many particles, each particle can have different numbers of degrees of freedom from the others. In each of the Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy.
Derivation of Lagrange's equations
Hamilton's principle
The Euler–Lagrange equations follow directly from Hamilton's principle, and are mathematically equivalent. From the calculus of variations, any functional of the form:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle J=\int_{x_1}^{x_2}F(x,y,y')\mathrm{d}x}
leads to the general Euler–Lagrange equation for stationary value of J. (see main article for derivation):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial F}{\partial y'}=\frac{\partial F}{\partial y}}
Then making the replacements:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x\rightarrow t,\quad y\rightarrow q,\quad y'\rightarrow \dot{q},\quad F\rightarrow L,\quad J\rightarrow\mathcal{S}}
yields the Lagrange equations for mechanics. Since mathematically Hamilton's equations can be derived from Lagrange's equations (by a Legendre transformation) and Lagrange's equations can be derived from Newton's laws, all of which are equivalent and summarize classical mechanics, this means classical mechanics is fundamentally ruled by a variation principle (Hamilton's principle above).
Generalized forces
For a conservative system, since the potential field is only a function of position, not velocity, Lagrange's equations also follow directly from the equation of motion above:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q_j = \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial \mathcal (L+V)}{\partial \dot{q}_j} \right )  \frac {\partial \mathcal (L+V)}{\partial q_j} = \left[\frac{\mathrm{d}}{\mathrm{d}t}\left ( \frac {\partial L}{\partial \dot{q}_j} \right ) +0\right]  \left[ \frac {\partial L}{\partial q_j}+\frac {\partial V}{\partial q_j}\right] = \frac{\mathrm{d}}{\mathrm{d}t}\left ( \frac {\partial L}{\partial \dot{q}_j} \right )  \frac {\partial L}{\partial q_j} + Q_j. }
simplifying to
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j} }
This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot{q}_j}
and time, and solely with respect to q_{j}, adding the results and associating terms with the equations for F_{i} and Q_{j}.
Newton's laws
As the following derivation shows, no new physics is introduced, so the Lagrange equations can describe the dynamics of a classical system equivalently as Newton's laws.
Derivation of Lagrange's equations from Newton's 2nd law and D'Alembert's principle

 Force and work done (on the particle)
Consider a single particle with mass m and position vector r, moving under an applied conservative force F, which can be expressed as the gradient of a scalar potential energy function V(r, t):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{F} =  \bold{\nabla} V. \, }
Such a force is independent of third or higherorder derivatives of r.
Consider an arbitrary displacement δr of the particle. The work done by the applied force F is
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \delta W = \bold{F} \cdot \delta \bold{r}.}
Using Newton's second law:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{F} \cdot \delta \bold{r} = m\ddot{\bold{r}} \cdot \delta \bold{r}. }
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,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bold{F} \cdot \bold{\delta} \bold{r} =  \bold{\nabla} V \cdot \displaystyle\sum_i {\partial \bold{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. }
On the right hand side, carrying out a change of coordinates to generalized coordinates, we obtain:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \ddot{r_i} {\partial r_i \over \partial q_j} \right] \delta q_j }
Now integrating by parts the summand with respect to t, then differentiating with respect to t:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\int\ddot{r_i} {\partial r_i \over \partial q_j} \mathrm{d}t = \frac{\mathrm{d}}{\mathrm{d}t}\left({\partial r_i \over \partial q_j}\dot{r}_i\right)\frac{\mathrm{d}}{\mathrm{d}t}\int\frac{\mathrm{d}}{\mathrm{d}t}\left({\partial r_i \over \partial q_j}\right)\dot{r}_i\mathrm{d}t= \frac{\mathrm{d}}{\mathrm{d}t}\left(\dot{r}_i{\partial r_i \over \partial q_j}\right)\dot{r}_i\frac{\mathrm{d}}{\mathrm{d}t}\left({\partial r_i \over \partial q_j}\right)}
allows the sum to be written as:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} \left( \dot{r_i} {\partial r_i \over \partial q_j} \right)  \dot{r_i} {\mathrm{d} \over \mathrm{d}t}\left( {\partial r_i \over \partial q_j} \right) \right] \right] \delta q_j }
Recognizing that
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\mathrm{d} \over \mathrm{d}t}{\partial r_j \over \partial q_i} = {\partial \dot{r_j} \over \partial q_i}, \quad {\partial r_j \over \partial q_i} = {\partial \dot{r_j} \over \partial \dot{q_i}},}
we obtain:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} \left( \dot{r_i} {\partial \dot{r_i} \over \partial \dot{q_j}} \right)  \dot{r_i} {\partial \dot{r_i} \over \partial q_j} \right] \right] \delta q_j }
 Kinetic and potential energy
Now, by changing the order of differentiation, we obtain:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{r} = m \sum_j \left[ \sum_i \left[ {\mathrm{d} \over \mathrm{d}t} {\partial \over \partial \dot{q_j}} \left( \frac{1}{2} \dot{r_i}^2 \right)  {\partial \over \partial q_j} \left( \frac{1}{2} \dot{r_i}^2 \right) \right] \right] \delta q_j }
Finally, we change the order of summation:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{r} = \sum_j \left[ {\mathrm{d} \over \mathrm{d}t} {\partial \over \partial \dot{q_j}} \left( \sum_i \frac{1}{2} m \dot{r_i}^2 \right)  {\partial \over \partial q_j} \left( \sum_i \frac{1}{2} m \dot{r_i}^2 \right) \right] \delta q_j }
Which is equivalent to:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m \ddot{\bold{r}} \cdot \delta \bold{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 is total kinetic energy of the system.
 Applying D'Alembert's principle
The equation for the work done becomes
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle m\mathbf{\ddot{r}}\cdot\delta \mathbf{r}\mathbf{F}\cdot\delta \mathbf{r}=\sum_i \left[{\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{\dot{q_i}}}{\partial{(TV)}\over \partial q_i}\right]\delta q_i = 0. }
However, this must be true for any set of generalized displacements δq_{i}, so we must have
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \left[ {\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{\dot{q_i}}}{\partial{(TV)}\over \partial q_i}\right] = 0 }
for each generalized coordinate δq_{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:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\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:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle {\partial{L}\over \partial q_i} = {\mathrm{d} \over \mathrm{d}t}{\partial{L}\over \partial{\dot{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.
Dissipation function
In a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the F_{i}, Rayleigh suggests using a dissipation function, D, of the following form:^{[7]}^{:271}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle D = \frac {1}{2} \sum_{j=1}^m \sum_{k=1}^m C_{j k} \dot{q}_j \dot{q}_k.}
where C_{jk} are constants that are related to the damping coefficients in the physical system, though not necessarily equal to them
If D is defined this way, then^{[7]}^{:271}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Q_j =  \frac {\partial V}{\partial q_j}  \frac {\partial D}{\partial \dot{q}_j}}
and
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 0 = \frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right )  \frac {\partial L}{\partial q_j} + \frac {\partial D}{\partial \dot{q}_j}.}
Examples
In this section two examples are provided in which the above concepts are applied. The first example establishes that in a simple case, the Newtonian approach and the Lagrangian formalism agree. The second case illustrates the power of the above formalism, in a case that is hard to solve with Newton's laws.
Falling mass
Consider a point mass m falling freely from rest. By gravity a force F = mg is exerted on the mass (assuming g constant during the motion). Filling in the force in Newton's law, we find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot x = g}
from which the solution
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle x(t) = \frac{1}{2} g t^2}
follows (by taking the antiderivative of the antiderivative, and choosing the origin as the starting point). This result can also be derived through the Lagrangian formalism. Take x to be the coordinate, which is 0 at the starting point. The kinetic energy is T = ^{1}⁄_{2}mv^{2} and the potential energy is V = −mgx; hence,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L = T  V = \frac{1}{2} m \dot{x}^2 + m g x.}
Then
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 0 = \frac{\partial L}{\partial x}  \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot x} = m g  m \frac{\mathrm{d} \dot x}{\mathrm{d} t} }
which can be rewritten as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot x = g}
, yielding the same result as earlier.
Pendulum on a movable support
Consider a pendulum of mass m and length ℓ, which is attached to a support with mass M, which can move along a line in the xdirection. Let x be the coordinate along the line of the support, and let us denote the position of the pendulum by the angle θ from the vertical.
Sketch of the situation with definition of the coordinates (click to enlarge)
The kinetic energy can then be shown to be
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \begin{array}{rcl} T &=& \frac{1}{2} M \dot{x}^2 + \frac{1}{2} m \left( \dot{x}_\mathrm{pend}^2 + \dot{y}_\mathrm{pend}^2 \right) \\ &=& \frac{1}{2} M \dot{x}^2 + \frac{1}{2} m \left[ \left( \dot x + \ell \dot\theta \cos \theta \right)^2 + \left( \ell \dot\theta \sin \theta \right)^2 \right], \end{array}}
and the potential energy of the system is
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle V = m g y_\mathrm{pend} =  m g \ell \cos \theta . }
The Lagrangian is therefore
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \begin{array}{rcl} L &=& T  V \\ &=& \frac{1}{2} M \dot{x}^2 + \frac{1}{2} m \left[ \left( \dot x + \ell \dot\theta \cos \theta \right)^2 + \left( \ell \dot\theta \sin \theta \right)^2 \right] + m g \ell \cos \theta \\ &=& \frac{1}{2} \left( M + m \right) \dot x^2 + m \dot x \ell \dot \theta \cos \theta + \frac{1}{2} m \ell^2 \dot \theta ^2 + m g \ell \cos \theta \end{array} }
Now carrying out the differentiations gives for the support coordinate x
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t} \left[ (M + m) \dot x + m \ell \dot\theta \cos\theta \right] = 0, }
therefore:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle (M + m) \ddot x + m \ell \ddot\theta\cos\thetam \ell \dot\theta ^2 \sin\theta = 0 }
indicating the presence of a constant of motion. Performing the same procedure for the variable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \theta}
yields:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}\left[ m( \dot x \ell \cos\theta + \ell^2 \dot\theta ) \right] + m \ell (\dot x \dot \theta + g) \sin\theta = 0;}
therefore
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot\theta + \frac{\ddot x}{\ell} \cos\theta + \frac{g}{\ell} \sin\theta = 0.\, }
These equations may look quite complicated, but finding them with Newton's laws would have required carefully identifying all forces, which would have been much more laborious and prone to errors. By considering limit cases, the correctness of this system can be verified: For example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot x \to 0}
should give the equations of motion for a pendulum that is at rest in some inertial frame, while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \ddot\theta \to 0}
should give the equations for a pendulum in a constantly accelerating system, etc. Furthermore, it is trivial to obtain the results numerically, given suitable starting conditions and a chosen time step, by stepping through the results iteratively.
Twobody central force problem
The basic problem is that of two bodies in orbit about each other attracted by a central force. The Jacobi coordinates are introduced; namely, the location of the center of mass R and the separation of the bodies r (the relative position). The Lagrangian is then^{[11]}^{[12]}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \begin{align} L &= TU = \frac {1}{2} M \dot{\mathbf{R}}^2 + \left( \frac {1}{2} \mu \dot{\mathbf{r}}^2  U(r) \right) \\ &= L_{\mathrm{cm}} + L_{\mathrm{rel}} \end{align}}
where M is the total mass, μ is the reduced mass, and U the potential of the radial force. The Lagrangian is divided into a centerofmass term and a relative motion term. The R equation from the Euler–Lagrange system is simply:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle M\ddot{\mathbf{R}} = 0, \, }
resulting in simple motion of the center of mass in a straight line at constant velocity. The relative motion is expressed in polar coordinates (r, θ):
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle L=\frac{1}{2} \mu \left(\dot r ^2 +r^2 \dot \theta ^2 \right)  U(r), }
which does not depend upon θ, therefore an ignorable coordinate. The Lagrange equation for θ is then:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac {\partial L}{\partial \dot \theta} = \mu r^2 \dot \theta = \mathrm{constant} = \ell, \, }
where ℓ is the conserved angular momentum. The Lagrange equation for r is:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \frac{\partial L}{\partial r} = \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot r}, \, }
or:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mu r \dot \theta ^2 \frac {dU}{dr} = \mu \ddot r. \, }
This equation is identical to the radial equation obtained using Newton's laws in a corotating reference frame, that is, a frame rotating with the reduced mass so it appears stationary. If the angular velocity is replaced by its value in terms of the angular momentum,
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \dot \theta = \frac {\ell}{\mu r^2}, \, }
the radial equation becomes:^{[13]}
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mu \ddot r = \frac{dU}{dr} + \frac{\ell^2}{\mu r^3}. \, }
which is the equation of motion for a onedimensional problem in which a particle of mass μ is subjected to the inward central force −dU/dr and a second outward force, called in this context the centrifugal force:
 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle F_{\mathrm{cf}} = \mu r \dot \theta ^2 = \frac {\ell^2}{\mu r^3}. \, }
Of course, if one remains entirely within the onedimensional formulation, ℓ enters only as some imposed parameter of the external outward force, and its interpretation as angular momentum depends upon the more general twodimensional problem from which the onedimensional problem originated.
If one arrives at this equation using Newtonian mechanics in a corotating frame, the interpretation is evident as the centrifugal force in that frame due to the rotation of the frame itself. If one arrives at this equation directly by using the generalized coordinates (r, θ) and simply following the Lagrangian formulation without thinking about frames at all, the interpretation is that the centrifugal force is an outgrowth of using polar coordinates. As Hildebrand says:^{[14]} "Since such quantities are not true physical forces, they are often called inertia forces. Their presence or absence depends, not upon the particular problem at hand, but upon the coordinate system chosen." In particular, if Cartesian coordinates are chosen, the centrifugal force disappears, and the formulation involves only the central force itself, which provides the centripetal force for a curved motion.
This viewpoint, that fictitious forces originate in the choice of coordinates, often is expressed by users of the Lagrangian method. This view arises naturally in the Lagrangian approach, because the frame of reference is (possibly unconsciously) selected by the choice of coordinates.^{[15]} Unfortunately, this usage of "inertial force" conflicts with the Newtonian idea of an inertial force. In the Newtonian view, an inertial force originates in the acceleration of the frame of observation (the fact that it is not an inertial frame of reference), not in the choice of coordinate system. To keep matters clear, it is safest to refer to the Lagrangian inertial forces as generalized inertial forces, to distinguish them from the Newtonian vector inertial forces. That is, one should avoid following Hildebrand when he says (p. 155) "we deal always with generalized forces, velocities accelerations, and momenta. For brevity, the adjective "generalized" will be omitted frequently."
It is known that the Lagrangian of a system is not unique. Within the Lagrangian formalism the Newtonian fictitious forces can be identified by the existence of alternative Lagrangians in which the fictitious forces disappear, sometimes found by exploiting the symmetry of the system.^{[16]}
Extensions of Lagrangian mechanics
The Hamiltonian, denoted by H, is obtained by performing a Legendre transformation on the Lagrangian, which introduces new variables, canonically conjugate to the original variables. This doubles the number of variables, but makes differential equations first order. 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 discovered 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.
Dissipation (i.e. nonconservative systems) can also be treated with an effective Lagrangian formulated by a certain doubling of the degrees of freedom; see.^{[17]}^{[18]}^{[19]}^{[20]}
See also
References
 ↑ Goldstein, H. (2001). Classical Mechanics (3rd ed.). AddisonWesley. p. 35.
 ↑ Goldstein, H. (2001). Classical Mechanics (3rd ed.). AddisonWesley. p. 54.
 ↑ ^{3.0} ^{3.1}
R. Dvorak, Florian Freistetter (2005). "§ 3.2 Lagrange equations of the first kind". Chaos and stability in planetary systems. Birkhäuser. p. 24. ISBN 3540282084.
 ↑
H Haken (2006). Information and selforganization (3rd ed.). Springer. p. 61. ISBN 3540330216.
 ↑
Cornelius Lanczos (1986). "II §5 Auxiliary conditions: the Lagrangian λmethod". The variational principles of mechanics (Reprint of University of Toronto 1970 4th ed.). Courier Dover. p. 43. ISBN 0486650677.
 ↑
Henry Zatzkis (1960). "§1.4 Lagrange equations of the second kind". In DH Menzel. Fundamental formulas of physics. 1 (2nd ed.). Courier Dover. p. 160. ISBN 0486605957.
 ↑ ^{7.0} ^{7.1} ^{7.2} ^{7.3} ^{7.4} ^{7.5} ^{7.6} ^{7.7} ^{7.8} Torby, Bruce (1984). "Energy Methods". Advanced Dynamics for Engineers. HRW Series in Mechanical Engineering. United States of America: CBS College Publishing. ISBN 0030633664.
 ↑ ^{8.0} ^{8.1} ^{8.2} Analytical Mechanics, L.N. Hand, J.D. Finch, Cambridge University Press, 2008, ISBN 9780521575720
 ↑ Torby1984, p.270
 ↑ The Road to Reality, Roger Penrose, Vintage books, 2007, ISBN 0679776311
 ↑ John Robert Taylor (2005). Classical mechanics. University Science Books. p. 297. ISBN 189138922X.
 ↑ The Lagrangian also can be written explicitly for a rotating frame. See Thanu Padmanabhan (2000). "§2.3.2 Motion in a rotating frame". Theoretical Astrophysics: Astrophysical processes (3rd ed.). Cambridge University Press. p. 48. ISBN 0521566320.
 ↑
Louis N. Hand, Janet D. Finch (1998). Analytical mechanics. Cambridge University Press. pp. 140–141. ISBN 0521575729.
 ↑
Francis Begnaud Hildebrand (1992). Methods of applied mathematics (Reprint of PrenticeHall 1965 2nd ed.). Courier Dover. p. 156. ISBN 0486670023.
 ↑
For example, see Michail Zak, Joseph P. Zbilut, Ronald E. Meyers (1997). From instability to intelligence. Springer. p. 202. ISBN 3540630554. for a comparison of Lagrangians in an inertial and in a noninertial frame of reference. See also the discussion of "total" and "updated" Lagrangian formulations in Ahmed A. Shabana (2008). Computational continuum mechanics. Cambridge University Press. pp. 118–119. ISBN 0521885698.
 ↑
Terry Gannon (2006). Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics. Cambridge University Press. p. 267. ISBN 0521835313.
 ↑ Kosyakov,, B. P. (2007). Introduction to the classical theory of particles and fields. Berlin, Germany: Springer. doi:10.1007/9783540409342.
 ↑ Galley, Chad R. (2013). "Classical Mechanics of Nonconservative Systems". Physical Review Letters. 110 (17): 174301. Bibcode:2013PhRvL.110q4301G. doi:10.1103/PhysRevLett.110.174301. PMID 23679733.
 ↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2014). "Radiation reaction at the level of the action". International Journal of Modern Physics A. 29 (24): 1450132. arXiv:1402.2610 . Bibcode:2014IJMPA..2950132B. doi:10.1142/S0217751X14501322.
 ↑ Birnholtz, Ofek; Hadar, Shahar; Kol, Barak (2013). "Theory of postNewtonian radiation and reaction". Physical Review D. 88 (10): 104037. Bibcode:2013PhRvD..88j4037B. doi:10.1103/PhysRevD.88.104037.
Further reading
 Landau, L.D. and Lifshitz, E.M. Mechanics, Pergamon Press.
 Gupta, Kiran Chandra, Classical mechanics of particles and rigid bodies (Wiley, 1988).
 Goldstein, Herbert, Classical Mechanics, Addison Wesley.
 Cassel, Kevin W.: Variational Methods with Applications in Science and Engineering, Cambridge University Press, 2013.
External links
