NationMaster - Encyclopedia: Lagrangian mechanics

Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by Joseph Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving Lagrange's equation, given herein, for each of the system's generalized coordinates. The fundamental lemma of calculus of variations shows that solving Lagrange's equation is equivalent to finding the path which minimizes the action functional, a quantity which is the integral of the Lagrangian over time. Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ... Look up conservation of energy in Wiktionary, the free dictionary. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... 1788 was a leap year starting on Tuesday (see link for calendar). ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... The fundamental lemma of the calculus of variations states that if f is a function in C [a,b], and for every function h âˆˆ C2[a,b] with h(a) = h(b) = 0, then f(x) is identically zero in the open interval (a,b). ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... This article is about the concept of integrals in calculus. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

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 time-varying 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. Look up Analysis in Wiktionary, the free dictionary An analysis is a critical evaluation, usually made by breaking a subject (either material or intellectual) down into its constituent parts, then describing the parts and their relationship to the whole. ... It has been suggested that this article or section be merged with Classical mechanics. ...

1 Lagrange's equations
- 1.1 Kinetic energy relations
2 Old Lagrange's equations
3 Examples
- 3.1 Falling mass
- 3.2 Pendulum on a movable support
4 Hamilton's principle
5 Extensions of Lagrangian mechanics
6 See also
7 References
8 Further reading
9 External links

Lagrange's equations

The equations of motion in Lagrangian mechanics are Lagrange's equations, also known as Euler-Lagrange equations. Below, we sketch out the derivation of Lagrange's equation. Please note that 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 Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. ...

Start with D'Alembert's principle for the virtual work of applied forces, $mathbf{F}_i$ , and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints:^[1]^:269 DAlembert DAlemberts-Lagrange principle is a statement of the fundamental classical laws of motion. ... A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. ... This article is about inertia as it applies to local motion. ...

$delta W = sum_{i=1}^n ( mathbf {F}_{i} - m_i mathbf{a}_i )cdot delta mathbf r_i = 0$ .

δ W

is the virtual work

$delta mathbf r_i$ is the virtual displacement of the system, consistent with the constraints

m i

are the masses of the particles in the system

$mathbf a_i$ are the accelerations of the particles in the system

$m_i mathbf a_i$ together as products represent the time derivatives of the system momenta, aka. inertial forces

i

is an integer used to indicate (via subscript) a variable corresponding to a particular particle

n

is the number of particles under consideration

Break out the two terms:

$delta W = sum_{i=1}^n mathbf {F}_{i} cdot delta mathbf r_i - sum_{i=1}^n m_i mathbf{a}_i cdot delta mathbf r_i = 0$ .

Assume that the following transformation equations from m independent generalized coordinates, $q j$ , hold:^[1]^:260 Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ...

$mathbf{r}_1=mathbf{r}_1(q_1, q_2, ..., q_m, t)$ ,

$mathbf{r}_2=mathbf{r}_2(q_1, q_2, ..., q_m, t)$ , ...

$mathbf{r}_n=mathbf{r}_n(q_1, q_2, ..., q_m, t)$ .

m

(without a subscript) indicates the total number generalized coordinates

An expression for the virtual displacement (differential), $delta mathbf{r}_i$ , of the system is^[1]^:264 The concept of a virtual displacement is meaningful only when discussing a physical system subject to contraints on its motion. ...

$delta mathbf{r}_i = sum_{j=1}^m frac {partial mathbf {r}_i} {partial q_j} delta q_j$ .

j

is an integer used to indicate (via subscript) a variable corresponding to a generalized coordinate

The applied forces may be expressed in the generalized coordinates as generalized forces, $Q j$ ,^[1]^:265 Generalized forces are defined via coordinate transformation of applied forces, , on a system of n particles, i. ...

$Q_j = sum_{i=1}^n mathbf {F}_{i} cdot frac {partial mathbf {r}_i} {partial q_j}$ .

Combining the equations for $δ W$ , $delta mathbf{r}_i$ , and $Q j$ yields the following result after pulling the sum out of the dot product in the second term:^[1]^:269

$delta W = sum_{j=1}^m Q_j delta q_j - sum_{j=1}^m sum_{i=1}^n m_i mathbf{a}_i cdot frac {partial mathbf {r}_i} {partial q_j} delta q_j = 0$ .

Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the kinetic energy leaves^[1]^:270

$delta W = sum_{j=1}^m Q_j delta q_j - sum_{j=1}^m left ( frac {d}{d t} left ( frac {partial T}{partial dot{q}_j} right ) - frac {partial T}{partial q_j} right ) delta q_j = 0$ .

In the above equation, $δ q j$ is arbitrary, though it is—by definition—consistent with the constraints. So the relation must hold term-wise:^[1]^:270

$Q_j = frac {d}{d t} left ( frac {partial T}{partial dot{q}_j} right ) - frac {partial T}{partial q_j}$ .

If the $mathbf F_i$ are conservative, they may be represented by a scalar potential field, $V$ :^[1]^{:266 & 270} It has been suggested that this article or section be merged with Potential. ...

$mathbf F_i = - nabla V Rightarrow Q_j = - sum_{i=1}^n nabla V cdot frac {partial mathbf {r}_i} {partial q_j} = - frac {partial V}{partial q_j}$ .

The previous result may be easier to see by recognizing that $V$ is a function of the $mathbf {r}_i$ , which are in turn functions of $q j$ , and then applying the chain rule to the derivative of $V$ with respect to $q j$ . In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

The definition of the Lagrangian is^[1]^:270 A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ...

$mathcal{L} = T - V$ .

Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows:^[1]^:270

$0 = frac {d}{d t} left ( frac {partial mathcal L}{partial dot{q}_j} right ) - frac {partial mathcal L}{partial q_j}$ .

In a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the $mathbf F_i$ , Rayleigh suggests using a dissipation function, $D$ , of the following form:^[1]^:271 For other uses, see Viscosity (disambiguation). ... John William Strutt, 3rd Baron Rayleigh (12 November 1842 â€“ 30 June 1919) was an English physicist who (with William Ramsay) discovered the element argon, an achievement that earned him the Nobel Prize for Physics in 1904. ...

$D = frac {1}{2} sum_{j=1}^m sum_{k=1}^m C_{j k} dot{q}_j dot{q}_k$ .

C j k

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^[1]^:271

$Q_j = - frac {partial V}{partial q_j} - frac {partial D}{partial dot{q}_j}$ and

$0 = frac {d}{d t} left ( frac {partial mathcal L}{partial dot{q}_j} right ) - frac {partial mathcal L}{partial q_j} + frac {partial D}{partial dot{q}_j}$ .

Kinetic energy relations

The kinetic energy, $T$ , for the system of particles is defined by^[1]^:269 The kinetic energy of an object is the extra energy which it possesses due to its motion. ...

$T = frac {1}{2} sum_{i=1}^n m_i mathbf {v}_i cdot mathbf {v}_i$ .

The partial derivative of $T$ with respect to the time derivatives of the generalized coordinates, $dot{q}_j$ , is^[1]^:269 A time derivative is a derivative of a function with respect to time, t. ...

$frac {partial T}{partial dot{q}_j} = sum_{i=1}^n m_i mathbf {v}_i cdot frac {partial mathbf {v}_i} {partial dot{q}_j}$ .

The previous result may be difficult to visualize. As a result of the product rule, the derivative of a general dot product d(f(x)•g(x))/dx is f(x)•dg(x)/dx + g(x)•df(x)/dx. The general result may be seen by briefly stepping into a Cartesian coordinate system, recognizing that the dot product is (there) a term-by-term product sum, and also recognizing that the derivative of a sum is the sum of its derivatives. In our case, f and g are equal to v, which is why the factor of one half disappears. In calculus, the product rule also called Leibnizs law (see derivation), governs the differentiation of products of differentiable functions. ... In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors over the real numbers R and returns a real-valued scalar quantity. ... Fig. ...

According to the chain rule and the coordinate transformation equations given above for $mathbf{r}$ , it's time derivative, $mathbf{v}$ , is:^[1]^:264 In calculus, the chain rule is a formula for the derivative of the composite of two functions. ...

$mathbf{v}_i = sum_{j=1}^m frac {partial mathbf{r}_i}{partial q_j} dot{q}_j + frac {partial mathbf{r}_i}{partial t}$ .

Together, the definition of $mathbf v_i$ and the total differential, $d mathbf {r}_i$ , suggest that^[1]^:269

$frac {partial mathbf {v}_i}{partial dot{q}_j} = frac {partial mathbf {r}_i}{partial q_j}$ .^[clarify]

Substituting this relation back into the expression for the partial derivative of $T$ gives^[1]^:269

$frac {partial T}{partial dot{q}_j} = sum_{i=1}^n m_i mathbf v_i cdot frac {partial mathbf {r}_i}{partial q_j}$ .

Taking the time derivative gives^[1]^:270

$frac {d}{d t} left ( frac {partial T}{partial dot{q}_j} right ) = sum_{i=1}^n left [ m_i mathbf a_i cdot frac {partial mathbf {r}_i}{partial q_j} + m_i mathbf {v}_i cdot frac {d}{d t} left ( frac {partial mathbf {r}_i}{partial q_j} right ) right ]$ .

Using the chain rule on the last term gives^[1]^:270

$frac {d}{d t} left ( frac {partial mathbf {r}_i}{partial q_j} right ) = sum_{k=1}^m frac {partial^2 mathbf r_i}{partial q_j partial q_k} dot{q_k} + frac {partial^2 mathbf r_i}{partial q_j partial t}$ .

From the expression for $mathbf v_i$ , one sees that^[1]^:270

$frac {d}{d t} left ( frac {partial mathbf {r}_i}{partial q_j} right ) = frac {partial mathbf {v}_i}{partial q_j}$ .

This allows simplification of the last term,^[1]^:270

$frac {d}{d t} left ( frac {partial T}{partial dot{q}_j} right ) = sum_{i=1}^n left [ m_i mathbf a_i cdot frac {partial mathbf {r}_i}{partial q_j} + m_i mathbf {v}_i cdot frac {partial mathbf {v}_i}{partial q_j} right ]$ .

The partial derivative of $T$ with respect to the generalized coordinates, $q j$ , is^[1]^:270

$frac {partial T}{partial q_j} = sum_{i=1}^n m_i mathbf {v}_i cdot frac {partial mathbf {v}_i} {partial q_j}$ .^[clarify]

The last two equations may be combined to give an expression for the inertial forces in terms of the kinetic energy:^[1]^:270

$frac {d}{d t} left ( frac {partial T}{partial dot{q}_j} right ) - frac {partial T}{partial q_j} = sum_{i=1}^n m_i mathbf a_i cdot frac {partial mathbf {r}_i}{partial q_j}$

Old Lagrange's equations

Consider a single particle with mass m and position vector $bold{r}$ , moving under an applied force, $bold{F}$ , which can be expressed as the gradient of a scalar potential energy function $V (bold{r},t)$ : For other uses, see Mass (disambiguation). ... A position vector is a vector used to describe the spatial position of a point relative to a reference point called the origin (of the space). ... For other uses, see Force (disambiguation). ... For other uses, see Gradient (disambiguation). ...

$bold{F} = - bold{nabla} V.$

Such a force is independent of third- or higher-order derivatives of $bold{r}$ , so Newton's second law forms a set of 3 second-order ordinary differential equations. Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom. An obvious set of variables is ${ bold{r}_j, dot{bold{r}}_j | j = 1, 2, 3}$ , the Cartesian components of $bold{r}$ and their time derivatives, at a given instant of time (i.e. position (x,y,z) and velocity $(v x, v y, v z)$ ). Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

More generally, we can work with a set of generalized coordinates, $q j$ , and their time derivatives, the generalized velocities, $dot{q_j}$ . The position vector, $bold{r}$ , is related to the generalized coordinates by some transformation equation: Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ... Generalized coordinates include any nonstandard coordinate system applied to the analysis of a physical system, especially in the study of Lagrangian dynamics. ...

$bold{r} = bold{r}(q_i , q_j , q_k, t).$

For example, for a simple pendulum of length l, a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be A gravity pendulum is a weight on the end of a rigid rod, which, when given some initial lift from the vertical position, will swing back and forth under the influence of gravity over its central (lowest) point. ...

$bold{r}(theta, dot{theta} , t) = (l sin theta, l cos theta)$ .

The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system. Fig. ...

Consider an arbitrary displacement $delta bold{r}$ of the particle. The work done by the applied force $bold{F}$ is $W = bold{F} cdot delta bold{r}$ . Using Newton's second law, we write: In physics, mechanical work is the amount of energy transferred by a force. ...

$begin{matrix} bold{F} cdot delta bold{r} = mddot{bold{r}} cdot delta bold{r}. end{matrix}$

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,

$begin{matrix} bold{F} cdot bold{delta} bold{r} & = & - bold{nabla} V cdot displaystylesum_i {partial bold{r} over partial q_i} delta q_i & = & - displaystylesum_{i,j} {partial V over partial r_j} {partial r_j over partial q_i} delta q_i & = & - displaystylesum_i {partial V over partial q_i} delta q_i. end{matrix}$

On the right hand side, carrying out a change of coordinates^[clarify], we obtain:

$m ddot{bold{r}} cdot delta bold{r} = m sum_{i,j} ddot{r_i} {partial r_i over partial q_j} delta q_j$

Rearranging Slightly:

$m ddot{bold{r}} cdot delta bold{r} = m sum_j left[ sum_i ddot{r_i} {partial r_i over partial q_j} right] delta q_j$

Now, by performing an "integration by parts" transformation, with respect to t:

$m ddot{bold{r}} cdot delta bold{r} = m sum_j left[ sum_i left[ {mathrm{d} over mathrm{d}t} left( dot{r_i} {partial r_i over partial q_j} right) - dot{r_i} {mathrm{d} over mathrm{d}t}left( {partial r_i over partial q_j} right) right] right] delta q_j$

Recognizing that ${mathrm{d} over mathrm{d}t}{partial r_j over partial q_i} = {partial dot{r_j} over partial q_i}$ and ${partial r_j over partial q_i} = {partial dot{r_j} over partial dot{q_i}}$ , we obtain:

$m ddot{bold{r}} cdot delta bold{r} = m sum_j left[ sum_i left[ {mathrm{d} over mathrm{d}t} left( dot{r_i} {partial dot{r_i} over partial dot{q_j}} right) - dot{r_i} {partial dot{r_i} over partial q_j} right] right] delta q_j$

Now, by changing the order of differentiation, we obtain:

$m ddot{bold{r}} cdot delta bold{r} = m sum_j left[ sum_i left[ {mathrm{d} over mathrm{d}t} {partial over partial dot{q_j}} left( frac{1}{2} dot{r_i}^2 right) - {partial over partial q_j} left( frac{1}{2} dot{r_i}^2 right) right] right] delta q_j$

Finally, we change the order of summation:

$m ddot{bold{r}} cdot delta bold{r} = sum_j left[ {mathrm{d} over mathrm{d}t} {partial over partial dot{q_j}} left( sum_i frac{1}{2} m dot{r_i}^2 right) - {partial over partial q_j} left( sum_i frac{1}{2} m dot{r_i}^2 right) right] delta q_j$

Which is equivalent to:

$m ddot{bold{r}} cdot delta bold{r} = sum_i left[{mathrm{d} over mathrm{d}t}{partial T over partial dot{q_i}}-{partial T over partial q_i}right]delta q_i$

where $T=frac{1}{2}mdot{bold{r}}cdotdot{bold{r}}$ is the kinetic energy of the particle. Our equation for the work done becomes

$sum_i left[{mathrm{d} over mathrm{d}t}{partial{T}over partial{dot{q_i}}}-{partial{(T-V)}over partial q_i}right] delta q_i = 0.$

However, this must be true for any set of generalized displacements $δ q i$ , so we must have

$left[ {mathrm{d} over mathrm{d}t}{partial{T}over partial{dot{q_i}}}-{partial{(T-V)}over partial q_i}right] = 0$

for each generalized coordinate $δ q i$ . 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:

${mathrm{d} over mathrm{d}t}{partial{V}over partial{dot{q_i}}} = 0.$

Inserting this into the preceding equation and substituting L = T - V, called the Lagrangian, we obtain Lagrange's equations:

${partial{L}over partial q_i} = {mathrm{d} over mathrm{d}t}{partial{L}over partial{dot{q_i}}}.$

There is one Lagrange equation for each generalized coordinate 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.

The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy.

In practice, it is often easier to solve a problem using the Euler-Lagrange equations than Newton's laws. This is because appropriate generalized coordinates q_i may be chosen to exploit symmetries in the system. The Euler-Lagrange equation, developed by Leonhard Euler and Joseph-Louis Lagrange in the 1750s, is the major formula of the calculus of variations. ...

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 which is hard to solve with Newton's laws.

Falling mass

Consider a point mass m falling freely from rest. By gravity a force F = m g is exerted on the mass (assuming g constant during the motion). Filling in the force in Newton's law, we find $ddot x = g$ from which the solution

$x(t) = frac{1}{2} g t^2$

follows (choosing the origin at the starting point). This result can also be derived through the Lagrange formalism. Take x to be the coordinate, which is 0 at the starting point. The kinetic energy is $T = frac{1}{2} m v^2$ and the potential energy is $V = - m g x$ , hence

$L = T - V = frac{1}{2} m dot{x}^2 + m g x$ .

Now we find

$0 = frac{partial L}{partial x} - frac{mathrm{d}}{mathrm{d}t} frac{partial L}{partial dot x} = m g - m frac{mathrm{d} dot x}{mathrm{d} t}$

which can be rewritten as $ddot x = g$ , 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 x-direction. 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. The kinetic energy can then be shown to be

$T = frac{1}{2} M dot{x}^2 + frac{1}{2} m left( dot{x}_mathrm{pend}^2 + dot{y}_mathrm{pend}^2 right) = frac{1}{2} M dot{x}^2 + frac{1}{2} m left[ left( dot x + l dottheta cos theta right)^2 + left( l dottheta sin theta right)^2 right],$

and the potential energy of the system is

$V = m g operatorname{y}_mathrm{pend} = - m g l cos theta .$

Sketch of the situation with definition of the coordinates (click to enlarge)

Now carrying out the differentiations gives for the support coordinate x Image File history File links PendulumWithMovableSupport. ... Image File history File links PendulumWithMovableSupport. ...

$frac{mathrm{d}}{mathrm{d}t} left[ (M + m) dot x + m l dottheta costheta right] = 0,$

therefore:

$(M + m) ddot x + m l ddotthetacostheta-m l dottheta ^2 sintheta = 0$

indicating the presence of a constant of motion. The other variable yields

$frac{mathrm{d}}{mathrm{d}t}left[ m( l^2 dottheta + dot x l costheta ) right] + m (dot x l dot theta + g l) sintheta = 0$ ;

therefore

$ddottheta + frac{ddot x}{l} costheta + frac{g}{l} sintheta = 0$ .

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 harder and prone to errors. By considering limit cases ( $ddot x to 0$ should give the equations of motion for a pendulum, $ddottheta to 0$ should give the equations for a pendulum in a constantly accelerating system, etc.) the correctness of this system can be verified.

Hamilton's principle

The action, denoted by $mathcal{S}$ , is the time integral of the Lagrangian:

$mathcal{S} = int L,mathrm{d}t.$

Let q₀ and q₁ be the coordinates at respective initial and final times t₀ and t₁. Using the calculus of variations, it can be shown the Lagrange's equations are equivalent to Hamilton's principle: Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ...

The system undergoes the trajectory between t₀ and t₁ whose action has a stationary value.

By stationary, we mean that the action does not vary to first-order for infinitesimal deformations of the trajectory, with the end-points (q₀, t₀) and (q₁,t₁) fixed. Hamilton's principle can be written as:

$delta mathcal{S} = 0. ,!$

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, this is a misnomer: the action only needs to be stationary, and the correct trajectory could be produced by a maximum, saddle point, or minimum in the action. The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... Plot of y = x3 with a saddle-point at (0,0). ...

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. Newtons laws of motion are the three scientific laws which Isaac Newton discovered concerning the behaviour of moving bodies. ... In mathematics, the term holonomic may occur with several different meanings. ... Jean le Rond dAlembert, pastel by Maurice Quentin de la Tour Jean Le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. ...

Extensions of Lagrangian mechanics

The Hamiltonian, denoted by H, is obtained by performing a Legendre transformation on the Lagrangian. 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)). Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Diagram illustrating the Legendre transformation of the function f(x) . The function is shown in red, and the tangent line at x0 is shown in blue. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

In 1948, Feynman invented 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. Year 1948 (MCMXLVIII) was a leap year starting on Thursday (link will display the 1948 calendar) of the Gregorian calendar. ... This article is about the physicist. ... This article or section is in need of attention from an expert on the subject. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... Properties The electron (also called negatron, commonly represented as e−) is a subatomic particle. ... In physics, the photon (from Greek Ï†Ï‰Ï‚, phÅs, meaning light) is the quantum of the electromagnetic field; for instance, light. ... Fermats principle assures that the angles given by Snells law always reflect lights quickest path between P and Q. Fermats principle in optics states: This principle was first stated by Pierre de Fermat. ... For the book by Sir Isaac Newton, see Opticks. ...

References

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q ^r ^s ^t ^u ^v ^w Torby, Bruce (1984). "Energy Methods", Advanced Dynamics for Engineers, HRW Series in Mechanical Engineering (in English). United States of America: CBS College Publishing. ISBN 0-03-063366-4.

Goldstein, H. Classical Mechanics, second edition, pp.16 (Addison-Wesley, 1980)
Moon, F. C. Applied Dynamics With Applications to Multibody and Mechatronic Systems, pp. 103-168 (Wiley, 1998).

External links

Tong, David, Classical Dynamics Cambridge lecture notes
Principle of least action interactive Excellent interactive explanation/webpage
Aerospace dynamics lecture notes on Lagrangian mechanics
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Lagrangian mechanics (934 words)
	Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.
	In Lagrangian mechanics, the trajectory of an object is derived by finding the path which minimizes the action, a quantity which is the integral of the Lagrangian over time.
	The Lagrangian for classical mechanics is taken to be the difference between the kinetic energy and the potential energy.

Lagrangian mechanics - Wikipedia, the free encyclopedia (926 words)
	Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788.
	In Lagrangian mechanics, the trajectory of an object is derived by finding the path which minimizes the action, a quantity which is the integral of the Lagrangian over time.
	The Lagrangian for classical mechanics is taken to be the difference between the kinetic energy and the potential energy.

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