Lagrangian mechanics

Classical mechanics

$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
Newton's Second Law

Formulations
Newtonian mechanics Lagrangian mechanics Hamiltonian mechanics

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Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may It was introduced by Joseph Louis Lagrange in 1788. Year 1788 ( MDCCLXXXVIII) was a Leap year starting on Tuesday (link will display the full calendar of the Gregorian calendar (or a Leap 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. By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately The fundamental lemma of calculus of variations shows that solving Lagrange's equation is equivalent to finding the path that minimizes the action functional, a quantity that is the integral of the Lagrangian over time. In Mathematics, specifically in the Calculus of variations, the fundamental lemma in the calculus of variations is a lemma that is typically used to transform In Physics, the action is a particular quantity in a Physical system that can be used to describe its operation The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system

The use of generalized coordinates may considerably simplify a system's analysis. Analysis (from Greek ἀνάλυσις, "a breaking up" is the process of breaking a complex topic or substance into smaller parts to gain a 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. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects 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.

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. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions 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.

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

$\delta W = \sum_{i=1}^n ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0$ . D'Alembert's principle, also known as the Lagrange-D'Alembert principle, is a statement of the fundamental classical laws of motion Virtual work on a system is the work resulting from either virtual forces acting through a real displacement or real Forces acting through a The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving

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

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

$\mathbf{r}_2=\mathbf{r}_2(q_1, q_2, ..., q_m, t)$ , . By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately . .

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

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

$\delta \mathbf{r}_i = \sum_{j=1}^m \frac {\partial \mathbf {r}_i} {\partial q_j} \delta q_j$ . A virtual displacement \delta \mathbf {r}_i "is an assumed infinitesimal change of system coordinates occurring while time is held constant

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

$Q_j = \sum_{i=1}^n \mathbf {F}_{i} \cdot \frac {\partial \mathbf {r}_i} {\partial q_j}$ . Generalized forces are defined via coordinate transformation of applied forces \mathbf{F}_i on a system of n particles i

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}

$\mathbf F_i = - \nabla V \Rightarrow Q_j = - \sum_{i=1}^n \nabla V \cdot \frac {\partial \mathbf {r}_i} {\partial q_j} = - \frac {\partial V}{\partial q_j}$ . A scalar Potential is a fundamental concept in Vector analysis and Physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion

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

$\mathcal{L} = T - V$ . The Lagrangian, L of a Dynamical system is a function that summarizes the dynamics of the system

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

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

This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to $\dot{q}_j$ and time, and solely with respect to $q j$ , adding the results and associating terms with the equations for $\mathbf F_i$ and $Q j$ .

In a more general formulation, the forces could be both potential and viscous. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. 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

$D = \frac {1}{2} \sum_{j=1}^m \sum_{k=1}^m C_{j k} \dot{q}_j \dot{q}_k$ . John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered

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

$T = \frac {1}{2} \sum_{i=1}^n m_i \mathbf {v}_i \cdot \mathbf {v}_i$ . The kinetic energy of an object is the extra Energy which it possesses due to its motion

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

$\frac {\partial T}{\partial \dot{q}_j} = \sum_{i=1}^n m_i \mathbf {v}_i \cdot \frac {\partial \mathbf {v}_i} {\partial \dot{q}_j}$ . A time derivative is a Derivative of a function with respect to Time, usually interpreted as the Rate of change of the value of the function

The previous result may be difficult to visualize. As a result of the product rule, the derivative of a general dot product $\frac{d}{dx} ( \mathbf{f}(x) \cdot \mathbf{g}(x) )$ is $\mathbf{f}(x) \cdot \frac{d}{dx} \mathbf{g}(x) + \mathbf{g}(x) \cdot \frac{d}{dx} \mathbf{f}(x)$ This 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 Calculus, the product rule also called Leibniz's law (see derivation) governs the differentiation of products of differentiable In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane In our case, f and g are equal to v, which is why the factor of one half disappears.

According to the chain rule and the coordinate transformation equations given above for $\mathbf{r}$ , it's time derivative, $\mathbf{v}$ , is:^[1]^:264

$\mathbf{v}_i = \sum_{j=1}^m \frac {\partial \mathbf{r}_i}{\partial q_j} \dot{q}_j + \frac {\partial \mathbf{r}_i}{\partial t}$ . In Calculus, the chain rule is a Formula for the Derivative of the composite of two functions.

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

$\frac {\partial \mathbf {v}_i}{\partial \dot{q}_j} = \frac {\partial \mathbf {r}_i}{\partial q_j}$ .

[ Remember that : $\frac {\partial } {\partial {\dot{q}_k}} {A}{\dot{q}_k} = A$ , and it is easier to visualise the result if you replace the subscript $j$ with some other subscript $k$ . Also remember that in the sum, there is only one ${\dot{q}_k}$ . ]

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

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

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

[This last result may be obtained by doing a partial differentiation directly on the kinetic energy definition represented by the first equation. ] 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 $V (\bold{r},t)$ :

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

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. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary reference In Physics, a force is whatever can cause an object with Mass to Accelerate. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the 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 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 $(v x, v y, v z)$ ).

More generally, we can work with a set of generalized coordinates, $q j$ , and their time derivatives, the generalized velocities, $\dot{q_j}$ . By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately The position vector, $\bold{r}$ , is related to the generalized coordinates by some transformation equation:

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

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

$\bold{r}(\theta, \dot{\theta} , t) = (l \sin \theta, l \cos \theta)$ . A pendulum is a mass that is attached to a pivot from which it can swing freely

The term "generalized coordinates" is really a holdover from the period when Cartesian coordinates were the default coordinate system. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

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

$\begin{matrix} \bold{F} \cdot \delta \bold{r} = m\ddot{\bold{r}} \cdot \delta \bold{r}. \end{matrix}$

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,

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

On the right hand side, carrying out a change of coordinates, 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 $T=\frac{1}{2}m\dot{\bold{r}}\cdot\dot{\bold{r}}$ 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:

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

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. In Calculus of variations, the Euler–Lagrange equation, or Lagrange's equation is a Differential equation whose solutions are the functions This is because appropriate generalized coordinates q_i may be chosen to exploit symmetries in the system.

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

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

Now we find

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

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

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

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

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

therefore:

$(M + m) \ddot x + m l \ddot\theta\cos\theta-m l \dot\theta ^2 \sin\theta = 0$

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

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

therefore

$\ddot\theta + \frac{\ddot x}{l} \cos\theta + \frac{g}{l} \sin\theta = 0$ .

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 harder and prone to errors. By considering limit cases ( $\ddot x \to 0$ should give the equations of motion for a pendulum, $\ddot\theta \to 0$ 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:

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

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:

The system undergoes the trajectory between t₀ and t₁ whose action has a stationary value. Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. In Physics, Hamilton's principle is William Rowan Hamilton 's formulation of the Principle of stationary action (see that article for historical formulations

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. This article discusses the history of 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. In Mathematics, a saddle point is a point in the domain of a function of two variables which is a Stationary point but not a Local extremum

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. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the 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. In Mathematics, the term holonomic may occur with several different meanings Virtual work on a system is the work resulting from either virtual forces acting through a real displacement or real Forces acting through a Working only with holonomic constraints is the price we have to pay for using an elegant variational formulation of mechanics.

Extensions of Lagrangian mechanics

The Hamiltonian, denoted by H, is obtained by performing a Legendre transformation on the Lagrangian. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics, it is often desirable to express a functional relationship f(x\ as a different function whose argument is the derivative of f rather The Hamiltonian is the basis for an alternative formulation of classical mechanics known as Hamiltonian mechanics. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. It is a particularly ubiquitous quantity in quantum mechanics (see Hamiltonian (quantum mechanics)). Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Quantum mechanics, the Hamiltonian H is the Observable corresponding to the Total energy of the system

In 1948, Feynman invented the path integral formulation extending the principle of least action to quantum mechanics for electrons and photons. Year 1948 ( MCMXLVIII) was a Leap year starting on Thursday (link will display the 1948 calendar of the Gregorian calendar. Richard Phillips Feynman (ˈfaɪnmən May 11 1918 – February 15 1988 was an American Physicist known for the Path integral formulation of quantum This article is about a formulation of quantum mechanics For integrals along a path also known as line or contour integrals see Line integral. This article discusses the history of the principle of least action Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena 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. In Optics, Fermat's principle or the principle of least time is the idea that the path taken between two points by a ray of light is the path that can be

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). In Physics and Astronomy, Euler's three-body problem is to solve for the motion of a particle that is acted upon by the Gravitational field of two other Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics and theoretical Physics, the functional derivative is a generalization of the Directional derivative. Lagrange's Equations in Lagrangian mechanics are usually written in the form {\mathrm{d} \over \mathrm{d}t}{\partial{T}\over \partial{q'_j}}-{\partial{T}\over In Mathematics and Classical mechanics, canonical coordinates are particular sets of coordinates on the Phase space, or equivalently on the Cotangent By deriving Equations of motion in terms of a general set of generalized coordinates, the results found will be valid for any Coordinate system that is ultimately Lagrangian analysis is application of Lagrangian mechanics to analysis of various phenomena "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
Aerospace dynamics lecture notes on Rayleigh dissipation function

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