The Neutron Wonders
The Neutron Wonders blog is also make you know about world,about science,and about all what you want.
Rabu, 18 Mei 2011
Lama Menunggu
Kita tunggu saja bagaimana hasilnya. Hasil itu akan menentukan nasib mereka untuk lolos ke tingkat provinsi.
Selasa, 08 Februari 2011
The Lagrange
Lagrangian Equation
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis 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,[1] which treat constraints explicitly as extra equations, often using Lagrange multipliers;[2][3] or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates.[1][4] 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 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.
The equations of motion in Lagrangian mechanics are the Lagrange equations, also known as the Euler–Lagrange equations. Below, we sketch out the derivation of the Lagrange equations of the second kind. 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, , and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints,[5]:269
where
δW is the virtual work;
is the virtual displacement of the system, consistent with the constraints;
mi are the masses of the particles in the system;
are the accelerations of the particles in the system;
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; and
n is the number of particles under consideration.
Break out the two terms:
Assume that the following transformation equations from m independent generalized coordinates, qj, hold:[5]:260
where m (without a subscript) indicates the total number of generalized coordinates. An expression for the virtual displacement (differential), of the system for time-independent constraints is[5]:264
where 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, Qj:[5]:265
Combining the equations for δW, , and Qj yields the following result after pulling the sum out of the dot product in the second term:[5]:269
Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the kinetic energy leaves[5]:270
In the above equation, δqj is arbitrary, though it is by definition consistent with the constraints. So the relation must hold term-wise:[5]:270
If the are conservative, they may be represented by a scalar potential field, V:[5]:266 & 270
The previous result may be easier to see by recognizing that V is a function of the , which are in turn functions of qj, and then applying the chain rule to the derivative of V with respect to qj.
Recall the definition of the Lagrangian is [5]:270
Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows:
This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to and time, and solely with respect to qj, adding the results and associating terms with the equations for
and Qj.
In a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the , Rayleigh suggests using a dissipation function, D, of the following form:[5]:271
where Cjk 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[5]:271
and
The equations of motion in Lagrangian mechanics are the Lagrange equations, also known as the Euler–Lagrange equations. Below, we sketch out the derivation of the Lagrange equations of the second kind. 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, , and inertial forces on a three dimensional accelerating system of n particles, i, whose motion is consistent with its constraints,[5]:269
where
δW is the virtual work;
is the virtual displacement of the system, consistent with the constraints;
mi are the masses of the particles in the system;
are the accelerations of the particles in the system;
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; and
n is the number of particles under consideration.
Break out the two terms:
Assume that the following transformation equations from m independent generalized coordinates, qj, hold:[5]:260
where m (without a subscript) indicates the total number of generalized coordinates. An expression for the virtual displacement (differential), of the system for time-independent constraints is[5]:264
where 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, Qj:[5]:265
Combining the equations for δW, , and Qj yields the following result after pulling the sum out of the dot product in the second term:[5]:269
Substituting in the result from the kinetic energy relations to change the inertial forces into a function of the kinetic energy leaves[5]:270
In the above equation, δqj is arbitrary, though it is by definition consistent with the constraints. So the relation must hold term-wise:[5]:270
If the are conservative, they may be represented by a scalar potential field, V:[5]:266 & 270
The previous result may be easier to see by recognizing that V is a function of the , which are in turn functions of qj, and then applying the chain rule to the derivative of V with respect to qj.
Recall the definition of the Lagrangian is [5]:270
Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows:
This is consistent with the results derived above and may be seen by differentiating the right side of the Lagrangian with respect to and time, and solely with respect to qj, adding the results and associating terms with the equations for
and Qj.
In a more general formulation, the forces could be both potential and viscous. If an appropriate transformation can be found from the , Rayleigh suggests using a dissipation function, D, of the following form:[5]:271
where Cjk 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[5]:271
and
Kinetic energy relations
The kinetic energy, T, for the system of particles is defined by[5]:269
The partial derivative of T with respect to the time derivatives of the generalized coordinates, , is[5]:269
The previous result may be difficult to visualize. As a result of the product rule, the derivative of a general dot product is
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 our case, and
are equal to
, which is why the factor of one half disappears.
According to the chain rule and the coordinate transformation equations given above for , its time derivative,
, is[5]:264
Together, the definition of and the total differential,
, suggest that[5]:269
since
and that in the sum, there is only one
Substituting this relation back into the expression for the partial derivative of T gives[5]:269
Taking the time derivative gives[5]:270
Using the chain rule on the last term gives[5]:270
From the expression for , one sees that[5]:270
This allows simplification of the last term,[5]:270
The partial derivative of T with respect to the generalized coordinates, qj, is[5]:270
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:[5]:270
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 = mg is exerted on the mass (assuming g constant during the motion). Filling in the force in Newton's law, we find from which the solution
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 = 1⁄2mv2 and the potential energy is V = −mgx; hence,
.
Then
which can be rewritten as , yielding the same result as earlier.