Hamilton's principle states that the true evolution of a system described by
generalized coordinates between two specified states and at two specified times and is a
stationary point (a point where the
variation is zero) of the
action functional \mathcal{S}[\mathbf{q}] \ \stackrel{\mathrm{def}}{=}\ \int_{t_1}^{t_2} L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)\, dt where L(\mathbf{q},\dot{\mathbf{q}},t) is the
Lagrangian function for the system. In other words, any
first-order perturbation of the true evolution results in (at most)
second-order changes in \mathcal{S}. The action \mathcal{S} is a
functional, i.e., something that takes as its input a
function and returns a single number, a
scalar. In terms of
functional analysis, Hamilton's principle states that the true evolution of a physical system is a solution of the functional equation {{Equation box 1 \frac{\delta \mathcal{S}}{\delta \mathbf{q}(t)}=0. That is, the system takes a path in configuration space for which the action is stationary, with fixed boundary conditions at the beginning and the end of the path.
Derivation from Newton's laws of motion Although Hamilton's principle can be viewed as a postulate effectively replacing Newton's laws of motion, it can also be derived from Newton's laws. Starting with
d'Alembert's principle: \sum_{i} ( \mathbf {F}_{i} - m_i \mathbf{a}_i )\cdot \delta \mathbf r_i = 0. Where i is the index of a mass, \mathbf {F}_i is applied force (excluding constraint forces), \textbf a_i is acceleration of the mass, and \delta \mathbf r_i is the virtual displacement of the i-th mass consistent with the constraints. The above equation holds true for all times t so a definite integral with respect to t must also be 0: \int_{t_1}^{t_2} \sum_{i} ( \mathbf {F}_{i} - m_i \ddot \mathbf{r}_i )\cdot \delta \mathbf r_i dt = 0. \delta \mathbf r_i, the sum, and the integral can all be distributed: \int_{t_1}^{t_2} \sum_i \mathbf{F}_i \cdot \delta \mathbf{r}_i dt - \int_{t_1}^{t_2}\sum_i m_i \ddot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i dt = 0 Looking at the second term we can use the
product rule to find the relation: \frac{d}{dt} (m_i \dot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i) = m_i \ddot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i + m_i \dot{\mathbf{r}}_i \cdot \frac{d}{dt}(\delta \mathbf{r}_i) Variation \delta and the time derivative \frac{d}{dt} commute (meaning \frac{d}{dt}\delta \mathbf{r} = \delta \dot{\mathbf{r}}), so we can rearrange this to isolate our acceleration term: m_i \ddot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i = \frac{d}{dt} (m_i \dot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i) - m_i \dot{\mathbf{r}}_i \cdot \delta \dot{\mathbf{r}}_i Substituting this into the integral we get: \int_{t_1}^{t_2} \sum_i m_i \ddot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i \, dt = \sum_i \left[ m_i \dot{\mathbf{r}}_i \cdot \delta \mathbf{r}_i \right]_{t_1}^{t_2} - \int_{t_1}^{t_2} \sum_i m_i \dot{\mathbf{r}}_i \cdot \delta \dot{\mathbf{r}}_i \, dt In Hamilton's principle, we define the variations such that they vanish at the endpoints: \delta \mathbf{r}_i(t_1) = \delta \mathbf{r}_i(t_2) = 0. This makes the first term on the right disappear completely. Also \delta(\dot \mathbf r^2)=2\dot \mathbf r \cdot \delta \dot \mathbf r so the remaining term is: -\int_{t_1}^{t_2} \delta \left( \sum_i \frac{1}{2} m_i \dot{\mathbf{r}}_i^2 \right) dt = - \int_{t_1}^{t_2} \delta T \, dt Where T is the total kinetic energy. Looking at the applied force term, if the forces are conservative, they can be derived from a potential energy V(\mathbf{r}_1, \mathbf{r}_2, \dots). The work done by these forces during a virtual displacement is: \sum_i \mathbf{F}_i \cdot \delta \mathbf{r}_i = -\sum_i \nabla_i V \cdot \delta \mathbf{r}_i = -\delta V Substituting everything back into the integral gives: \int_{t_1}^{t_2} (-\delta V - (-\delta T)) \, dt = 0 \delta \int_{t_1}^{t_2} (T - V) \, dt = 0 Here the Lagrangian L=T-V. This is Hamilton's principle for a non-relativistic system of masses in the absence of an electromagnetic field.
Euler–Lagrange equations derived from the action integral Requiring that the true trajectory be a
stationary point of the action functional \mathcal{S} is equivalent to a set of differential equations for (the
Euler–Lagrange equations), which may be derived as follows. Let represent the true evolution of the system between two specified states and at two specified times and , and let be a small perturbation that is zero at the endpoints of the trajectory \boldsymbol\varepsilon(t_1) = \boldsymbol\varepsilon(t_2) \ \stackrel{\mathrm{def}}{=}\ 0 To first order in the perturbation , the change in the action functional \delta\mathcal{S} would be \begin{align} \delta \mathcal{S} &= \int_{t_1}^{t_2}\; \left[ L(\mathbf{q}+\boldsymbol\varepsilon,\dot{\mathbf{q}} +\dot{\boldsymbol{\varepsilon}})- L(\mathbf{q},\dot{\mathbf{q}}) \right]dt \\ &= \int_{t_1}^{t_2}\; \left( \boldsymbol\varepsilon \cdot \frac{\partial L}{\partial \mathbf{q}} + \dot{\boldsymbol{\varepsilon}} \cdot \frac{\partial L}{\partial \dot{\mathbf{q}}} \right)\,dt \end{align} where we have expanded the
Lagrangian L to first order in the perturbation . Applying
integration by parts to the last term results in \delta \mathcal{S} = \left[ \boldsymbol\varepsilon \cdot \frac{\partial L}{\partial \dot{\mathbf{q}}}\right]_{t_1}^{t_2} + \int_{t_1}^{t_2}\; \left( \boldsymbol\varepsilon \cdot \frac{\partial L}{\partial \mathbf{q}} - \boldsymbol\varepsilon \cdot \frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} \right)\,dt The boundary conditions \boldsymbol\varepsilon(t_1) = \boldsymbol\varepsilon(t_2) \ \stackrel{\mathrm{def}}{=}\ 0 causes the first term to vanish \delta \mathcal{S} = \int_{t_1}^{t_2}\; \boldsymbol\varepsilon \cdot\left(\frac{\partial L}{\partial \mathbf{q}} - \frac{d}{dt} \frac{\partial L}{\partial \dot{\mathbf{q}}} \right)\,dt Hamilton's principle requires that this first-order change \delta \mathcal{S} is zero for all possible perturbations , i.e., the true path is a
stationary point of the action functional \mathcal{S} (either a minimum, maximum or saddle point). This requirement can be satisfied if and only if {{Equation box 1 These equations are called the Euler–Lagrange equations for the variational problem.
Example: Free particle in polar coordinates Trivial examples help to appreciate the use of the action principle via the Euler–Lagrange equations. A free particle (mass
m and velocity
v) in Euclidean space moves in a straight line. Using the Euler–Lagrange equations, this can be shown in
polar coordinates as follows. In the absence of a potential, the Lagrangian is simply equal to the kinetic energy L = \frac{1}{2} mv^2= \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2 \right) in orthonormal (
x,
y) coordinates, where the dot represents differentiation with respect to the curve parameter (usually the time,
t). Therefore, upon application of the Euler–Lagrange equations, \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) - \frac{\partial L}{\partial x} = 0 \qquad \Rightarrow \qquad m\ddot{x} = 0 And likewise for
y. Thus the Euler–Lagrange formulation can be used to derive Newton's laws. In polar coordinates the kinetic energy and hence the Lagrangian becomes L = \frac{1}{2}m \left( \dot{r}^2 + r^2\dot{\varphi}^2 \right). The radial and components of the Euler–Lagrange equations become, respectively \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{r}} \right) - \frac{\partial L}{\partial r} = 0 \qquad \Rightarrow \qquad \ddot{r} - r\dot{\varphi}^2 = 0 \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\varphi}} \right)-\frac{\partial L}{\partial \varphi} = 0 \qquad \Rightarrow \qquad \ddot{\varphi} + \frac{2}{r}\dot{r}\dot{\varphi} = 0. remembering that r is also dependent on time and the product rule is needed to compute the total time derivative \frac{d}{dt} mr^2 \dot{\varphi}. The solution of these two equations is given by r = \sqrt{(a t + b)^2 + c^2} \varphi = \tan^{-1} \left( \frac{a t + b}{c} \right) + d for a set of constants , , , determined by initial conditions. Thus, indeed,
the solution is a straight line given in polar coordinates: is the velocity, is the distance of the closest approach to the origin, and is the angle of motion. ==Applied to deformable bodies==