Newton's laws (1642–1727) For simplicity, Newton's laws can be illustrated for one particle without much loss of generality (for a system of
N particles, all of these equations apply to each particle in the system). The
equation of motion for a particle of constant mass
m is
Newton's second law of 1687, in modern vector notation \mathbf{F} = m \mathbf{a}, where
a is its acceleration and
F the resultant force acting
on it. Where the mass is varying, the equation needs to be generalised to take the time derivative of the momentum. In three spatial dimensions, this is a system of three coupled second-order
ordinary differential equations to solve, since there are three components in this vector equation. The solution is the position vector
r of the particle at time
t, subject to the
initial conditions of
r and
v when Newton's laws are easy to use in Cartesian coordinates, but Cartesian coordinates are not always convenient, and for other coordinate systems the equations of motion can become complicated. In a set of
curvilinear coordinates the law in
tensor index notation is the
"Lagrangian form" F^a = m \left( \frac{\mathrm{d}^2 \xi^a}{\mathrm{d}t^2} + \Gamma^a{}_{bc} \frac{\mathrm{d}\xi^b}{\mathrm{d}t}\frac{\mathrm{d}\xi^c}{\mathrm{d}t} \right) = g^{ak} \left(\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial T}{\partial \dot{\xi}^k} - \frac{\partial T}{\partial \xi^k}\right), \quad \dot{\xi}^a \equiv \frac{\mathrm{d} \xi^a }{\mathrm{d}t}, where
Fa is the
a-th
contravariant component of the resultant force acting on the particle, Γ
abc are the
Christoffel symbols of the second kind, T = \frac{1}{2} m g_{bc} \frac{\mathrm{d} \xi^b}{\mathrm{d}t} \frac{\mathrm{d} \xi^c}{\mathrm{d}t} is the kinetic energy of the particle, and
gbc the
covariant components of the
metric tensor of the curvilinear coordinate system. All the indices
a,
b,
c, each take the values 1, 2, 3. Curvilinear coordinates are not the same as generalized coordinates. It may seem like an overcomplication to cast Newton's law in this form, but there are advantages. The acceleration components in terms of the Christoffel symbols can be avoided by evaluating derivatives of the kinetic energy instead. If there is no resultant force acting on the particle, it does not accelerate, but moves with constant velocity in a straight line. Mathematically, the solutions of the differential equation are
geodesics, the curves of extremal length between two points in space (these may end up being minimal, that is the shortest paths, but not necessarily). In flat 3D real space the geodesics are simply straight lines. So for a free particle, Newton's second law coincides with the geodesic equation and states that free particles follow geodesics, the extremal trajectories it can move along. If the particle is subject to forces the particle accelerates due to forces acting on it and deviates away from the geodesics it would follow if free. With appropriate extensions of the quantities given here in flat 3D space to 4D
curved spacetime, the above form of Newton's law also carries over to
Einstein's
general relativity, in which case free particles follow geodesics in curved spacetime that are no longer "straight lines" in the ordinary sense. However, we still need to know the total resultant force
F acting on the particle, which in turn requires the resultant non-constraint force
N plus the resultant constraint force
C, \mathbf{F} = \mathbf{C} + \mathbf{N}. The constraint forces can be complicated, since they generally depend on time. Also, if there are constraints, the curvilinear coordinates are not independent but related by one or more constraint equations. The constraint forces can either be eliminated from the equations of motion, so only the non-constraint forces remain, or included by including the constraint equations in the equations of motion.
D'Alembert's principle (1717–1783) {{multiple image A fundamental result in
analytical mechanics is
D'Alembert's principle, introduced in 1708 by
Jacques Bernoulli to understand
static equilibrium, and developed by
D'Alembert in 1743 to solve dynamical problems. The principle asserts for
N particles the virtual work, i.e. the work along a virtual displacement, \delta \mathbf{r}_k, is zero: i.e. in such a way that the constraint forces maintain the constrained motion. They are not the same as the actual displacements in the system, which are caused by the resultant constraint and non-constraint forces acting on the particle to accelerate and move it.
Virtual work is the work done along a virtual displacement for any force (constraint or non-constraint). Since the constraint forces act perpendicular to the motion of each particle in the system to maintain the constraints, the total virtual work by the constraint forces acting on the system is zero: \sum_{k=1}^N \mathbf {C}_k \cdot \delta \mathbf{r}_k = 0, so that \sum_{k=1}^N (\mathbf {N}_k - m_k \mathbf{a}_k ) \cdot \delta \mathbf{r}_k = 0. Thus D'Alembert's principle allows us to concentrate on only the applied non-constraint forces, and exclude the constraint forces in the equations of motion. The form shown is also independent of the choice of coordinates. However, it cannot be readily used to set up the equations of motion in an arbitrary coordinate system since the displacements \delta \mathbf{r}_k might be connected by a constraint equation, which prevents us from setting the
N individual summands to 0. We will therefore seek a system of mutually independent coordinates for which the total sum will be 0 if and only if the individual summands are 0. Setting each of the summands to 0 will eventually give us our separated equations of motion.
Equations of motion from D'Alembert's principle If there are constraints on particle
k, then since the coordinates of the position are linked together by a constraint equation, so are those of the
virtual displacements . Since the generalized coordinates are independent, we can avoid the complications with the
δrk by converting to virtual displacements in the generalized coordinates. These are related in the same form as a
total differential, or the
generalized equations of motion, Q_j = \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j} These equations are equivalent to Newton's laws
for the non-constraint forces. The generalized forces in this equation are derived from the non-constraint forces only – the constraint forces have been excluded from D'Alembert's principle and do not need to be found. The generalized forces may be non-conservative, provided they satisfy D'Alembert's principle.
Euler–Lagrange equations and Hamilton's principle (only some are shown). The path taken by the system (red) has a stationary action (δ
S = 0) under small changes in the configuration of the system (δ
q). For a non-conservative force which depends on velocity, it
may be possible to find a potential energy function
V that depends on positions and velocities. If the generalized forces
Qi can be derived from a potential
V such that Q_j = \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial V}{\partial \dot{q}_j} - \frac{\partial V}{\partial q_j}, equating to Lagrange's equations and defining the Lagrangian as obtains '''Lagrange's equations of the second kind
or the Euler–Lagrange equations''' of motion \frac{\partial L}{\partial q_j} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_j} = 0. However, the Euler–Lagrange equations can only account for non-conservative forces
if a potential can be found as shown. This may not always be possible for non-conservative forces, and Lagrange's equations do not involve any potential, only generalized forces; therefore they are more general than the Euler–Lagrange equations. The Euler–Lagrange equations also follow from the
calculus of variations. The
variation of the Lagrangian is \delta L = \sum_{j=1}^n \left(\frac{\partial L}{\partial q_j} \delta q_j + \frac{\partial L}{\partial \dot{q}_j} \delta \dot{q}_j \right),\quad \delta \dot{q}_j \equiv \delta\frac{\mathrm{d}q_j}{\mathrm{d}t} \equiv \frac{\mathrm{d}(\delta q_j)}{\mathrm{d}t}, which has a form similar to the
total differential of
L, but the virtual displacements and their time derivatives replace differentials, and there is no time increment in accordance with the definition of the virtual displacements. An
integration by parts with respect to time can transfer the time derivative of
δqj to the ∂
L/∂(d
qj/d
t), in the process exchanging d(
δqj)/d
t for
δqj, allowing the independent virtual displacements to be factorized from the derivatives of the Lagrangian, \begin{align} \int_{t_1}^{t_2} \delta L \, \mathrm{d}t & =\int_{t_1}^{t_2} \sum_{j=1}^n \left(\frac{\partial L}{\partial q_j}\delta q_j +\frac{\mathrm{d}}{\mathrm{d}t} \left(\frac{\partial L}{\partial \dot{q}_j}\delta q_j\right) - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_j}\delta q_j \right) \, \mathrm{d}t \\ &= \sum_{j=1}^n\left[\frac{\partial L}{\partial \dot{q}_j}\delta q_j\right]_{t_1}^{t_2} + \int_{t_1}^{t_2} \sum_{j=1}^n \left(\frac{\partial L}{\partial q_j} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{q}_j} \right)\delta q_j \, \mathrm{d}t. \end{align} Now, if the condition holds for all
j, the terms not integrated are zero. If in addition the entire time integral of
δL is zero, then because the
δqj are independent, and the only way for a definite integral to be zero is if the integrand equals zero, each of the coefficients of
δqj must also be zero. Then we obtain the equations of motion. This can be summarized by '''
Hamilton's principle''': \int_{t_1}^{t_2}\delta L \, \mathrm{d}t = 0. The time integral of the Lagrangian is another quantity called the
action, defined as S = \int_{t_1}^{t_2} L\,\mathrm{d}t, which is a
functional; it takes in the Lagrangian function for all times between
t1 and
t2 and returns a scalar value. Its dimensions are the same as , ·, or ·. With this definition Hamilton's principle is \delta S = 0. Instead of thinking about particles accelerating in response to applied forces, one might think of them picking out the path with a stationary action, with the end points of the path in configuration space held fixed at the initial and final times. Hamilton's principle is one of several
action principles. Historically, the idea of finding the shortest path a particle can follow subject to a force motivated the first applications of the
calculus of variations to mechanical problems, such as the
Brachistochrone problem solved by
Jean Bernoulli in 1696, as well as
Leibniz,
Daniel Bernoulli,
L'Hôpital around the same time, and
Newton the following year. Newton himself was thinking along the lines of the variational calculus, but did not publish. This will not be given here.
Lagrange multipliers and constraints The Lagrangian
L can be varied in the Cartesian
rk coordinates, for
N particles, \int_{t_1}^{t_2} \sum_{k=1}^N \left(\frac{\partial L}{\partial \mathbf{r}_k} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{\mathbf{r}}_k} \right)\cdot\delta \mathbf{r}_k \, \mathrm{d}t = 0. Hamilton's principle is still valid even if the coordinates
L is expressed in are not independent, here
rk, but the constraints are still assumed to be holonomic. As always the end points are fixed for all
k. What cannot be done is to simply equate the coefficients of
δrk to zero because the
δrk are not independent. Instead, the method of
Lagrange multipliers can be used to include the constraints. Multiplying each constraint equation by a Lagrange multiplier
λi for
i = 1, 2, ...,
C, and adding the results to the original Lagrangian, gives the new Lagrangian L' = L(\mathbf{r}_1,\mathbf{r}_2,\ldots,\dot{\mathbf{r}}_1,\dot{\mathbf{r}}_2,\ldots,t) + \sum_{i=1}^C \lambda_i(t) f_i(\mathbf{r}_k,t). The Lagrange multipliers are arbitrary functions of time
t, but not functions of the coordinates
rk, so the multipliers are on equal footing with the position coordinates. Varying this new Lagrangian and integrating with respect to time gives \int_{t_1}^{t_2} \delta L' \mathrm{d}t = \int_{t_1}^{t_2} \sum_{k=1}^N \left(\frac{\partial L}{\partial \mathbf{r}_k} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{\mathbf{r}}_k} + \sum_{i=1}^C \lambda_i \frac{\partial f_i }{\partial \mathbf{r}_k}\right)\cdot\delta \mathbf{r}_k \, \mathrm{d}t = 0. The introduced multipliers can be found so that the coefficients of
δrk are zero, even though the
rk are not independent. The equations of motion follow. From the preceding analysis, obtaining the solution to this integral is equivalent to the statement \frac{\partial L'}{\partial \mathbf{r}_k} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L'}{\partial \dot{\mathbf{r}}_k} = 0 \quad \Rightarrow \quad \frac{\partial L}{\partial \mathbf{r}_k} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \dot{\mathbf{r}}_k} + \sum_{i=1}^C \lambda_i \frac{\partial f_i }{\partial \mathbf{r}_k} = 0, which are '''Lagrange's equations of the first kind'
. Also, the λ''
i Euler-Lagrange equations for the new Lagrangian return the constraint equations \frac{\partial L'}{\partial \lambda_i} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L'}{\partial \dot{\lambda}_i} = 0 \quad \Rightarrow \quad f_i(\mathbf{r}_k,t) = 0. For the case of a conservative force given by the gradient of some potential energy
V, a function of the
rk coordinates only, substituting the Lagrangian gives \underbrace{ \frac{\partial T}{\partial \mathbf{r}_k} - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{\mathbf{r}}_k} }_{-\mathbf{F}_k} + \underbrace{ - \frac{\partial V}{\partial \mathbf{r}_k}}_{\mathbf{N}_k} + \sum_{i=1}^C \lambda_i \frac{\partial f_i}{\partial \mathbf{r}_k} = 0, and identifying the derivatives of kinetic energy as the (negative of the) resultant force, and the derivatives of the potential equaling the non-constraint force, it follows the constraint forces are \mathbf{C}_k = \sum_{i=1}^C \lambda_i \frac{\partial f_i}{\partial \mathbf{r}_k}, thus giving the constraint forces explicitly in terms of the constraint equations and the Lagrange multipliers. == Properties of the Lagrangian ==